**Authors**
David Joyner
David Phillips
Minh Van Nguyen

**License**
GFDL-1.3-no-invariants-or-later

Algorithmic Graph Theory and Sage David Joyner, Minh Van Nguyen, David Phillips Version 0.8-r1991 2013 May 10 Copyright © 2010–2013 David Joyner hwdjoyner@gmail.comi Copyright © 2009–2013 Minh Van Nguyen hmvngu.name@gmail.comi Copyright © 2013 David Phillips hdphillip@usna.edui Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. The latest version of the book is available from its website at http://code.google.com/p/graphbook/ Edition Version 0.8-r1991 2013 May 10 Contents Acknowledgments iv 1 Introduction to graph theory 1 1.1 Graphs and digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Subgraphs and other graph types . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Representing graphs in a computer . . . . . . . . . . . . . . . . . . . . . 19 1.4 Graph transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5 Isomorphic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6 New graphs from old . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 Trees and forests 51 2.1 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2 Properties of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3 Minimum spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.4 Binary trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.5 Huffman codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.6 Tree traversals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3 Shortest paths algorithms 103 3.1 Representing graphs in a computer . . . . . . . . . . . . . . . . . . . . . 104 3.2 Graph searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 3.3 Weights and distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 3.4 Dijkstra’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 3.5 Bellman-Ford algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.6 Floyd-Roy-Warshall algorithm . . . . . . . . . . . . . . . . . . . . . . . . 128 3.7 Johnson’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.8 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 4 Graph data structures 149 4.1 Priority queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2 Binary heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.3 Binomial heaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.4 Binary search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 i ii Contents 5 Distance and connectivity 182 5.1 Paths and distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.2 Vertex and edge connectivity . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3 Menger’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 5.4 Whitney’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.5 Centrality of a vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.6 Network reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.7 The spectrum of a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 5.8 Expander graphs and Ramanujan graphs . . . . . . . . . . . . . . . . . . 203 5.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Centrality and prestige 209 6.1 Vertex centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.2 Edge centrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.3 Ranking web pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.4 Hub and authority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7 Optimal graph traversals 210 7.1 Eulerian graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.2 Hamiltonian graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.3 The Chinese Postman Problem . . . . . . . . . . . . . . . . . . . . . . . 211 7.4 The Traveling Salesman Problem . . . . . . . . . . . . . . . . . . . . . . 211 8 Graph coloring 212 8.1 Vertex coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.2 Edge coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 8.3 The chromatic polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.4 Applications of graph coloring . . . . . . . . . . . . . . . . . . . . . . . . 221 9 Network flows 223 9.1 Flows and cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.2 Chip firing games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 9.3 Ford-Fulkerson theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.4 Edmonds and Karp’s algorithm . . . . . . . . . . . . . . . . . . . . . . . 240 9.5 Goldberg and Tarjan’s algorithm . . . . . . . . . . . . . . . . . . . . . . 240 10 Algebraic graph theory 241 10.1 Laplacian and adjacency matrices . . . . . . . . . . . . . . . . . . . . . . 241 10.2 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.3 Algebraic connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.4 Graph invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.5 Cycle and cut spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 10.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 11 Random graphs 242 11.1 Network statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 11.2 Binomial random graph model . . . . . . . . . . . . . . . . . . . . . . . . 246 11.3 Erdős-Rényi model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Contents iii 11.4 Small-world networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 11.5 Scale-free networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 11.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 A Asymptotic growth 271 B GNU Free Documentation License 272 1. APPLICABILITY AND DEFINITIONS . . . . . . . . . . . . . . . . . . . . 272 2. VERBATIM COPYING . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 3. COPYING IN QUANTITY . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 4. MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 5. COMBINING DOCUMENTS . . . . . . . . . . . . . . . . . . . . . . . . . 276 6. COLLECTIONS OF DOCUMENTS . . . . . . . . . . . . . . . . . . . . . . 277 7. AGGREGATION WITH INDEPENDENT WORKS . . . . . . . . . . . . . 277 8. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9. TERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 10. FUTURE REVISIONS OF THIS LICENSE . . . . . . . . . . . . . . . . . 278 11. RELICENSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 ADDENDUM: How to use this License for your documents . . . . . . . . . . . 279 Bibliography 280 Index 288 Acknowledgments Fidel Barrera-Cruz: reported typos in Chapter 2. See changeset 101. Suggested making a note about disregarding the direction of edges in undirected graphs. See changeset 277. Daniel Black: reported a typo in Chapter 1. See changeset 61. Kevin Brintnall: reported typos in the definition of iadj(v) ∩ oadj(v); see change- sets 240 and 242. Solution to Example 1.14(2); see changeset 246. John Costella: helped to clarify the idea that the adjacency matrix of a bipar- tite graph can be permuted to obtain a block diagonal matrix. See page 22 and revisions 1865 and 1869. Aaron Dutle: reported a typo in Figure 1.18. See changeset 125. Péter L. Erdős (http://www.renyi.hu/∼elp) for informing us of the reference [74] on the Havel-Hakimi theorem for directed graphs. Noel Markham: reported a typo in Algorithm 3.5. See changeset 131 and Issue 2. Caroline Melles: clarify definitions of various graph types (weighted graphs, multi- graphs, and weighted multigraphs); clarify definitions of degree, isolated vertices, and pendant and using the butterfly graph with 5 vertices (see Figure 1.10) to illustrate these definitions; clarify definitions of trails, closed paths, and cycles; see changeset 448. Some rearrangements of materials in Chapter 1 to make the reading flow better and a few additions of missing definitions; see changeset 584. Clarifications about unweighted and weighted degree of a vertex in a multigraph; notational convention about a graph being simple unless otherwise stated; an exam- ple on graph minor; see changeset 617. Reported a missing edge in Figure 1.5(b); see changeset 1945. Pravin Paratey: simplify the sentence formation in the definition of digraphs; see changeset 714 and Issue 7. Henrique Rennó: pointed out the ambiguity in the definition of weighted multi- graphs; see changeset 1936. Reported typos; see changeset 1938. The world map in Figure ?? was adapted from an SVG image file from Wikipedia. The original SVG file was accessed on 2010-10-01 at http://en.wikipedia.org/wiki/ File:WorldmapwlocationwNEDw50m.svg. iv Chapter 1 Introduction to graph theory — Spiked Math, http://spikedmath.com/120.html Our journey into graph theory starts with a puzzle that was solved over 250 years ago by Leonhard Euler (1707–1783). The Pregel River flowed through the town of Königsberg, which is present day Kaliningrad in Russia. Two islands protruded from the river. On either side of the mainland, two bridges joined one side of the mainland with one island and a third bridge joined the same side of the mainland with the other island. A bridge connected the two islands. In total, seven bridges connected the two islands with both sides of the mainland. A popular exercise among the citizens of Königsberg was determining if it was possible to cross each bridge exactly once during a single walk. For historical perspectives on this puzzle and Euler’s solution, see Gribkovskaia et al. [90] and Hopkins and Wilson [102]. To visualize this puzzle in a slightly different way, consider Figure 1.1. Imagine that points a and c are either sides of the mainland, with points b and d being the two islands. Place the tip of your pencil on any of the points a, b, c, d. Can you trace all the lines in the figure exactly once, without lifting your pencil? Known as the seven bridges of Königsberg puzzle, Euler solved this problem in 1735 and with his solution he laid the foundation of what is now known as graph theory. 1 2 Chapter 1. Introduction to graph theory c b d a Figure 1.1: The seven bridges of Königsberg puzzle. 1.1 Graphs and digraphs When I use a word, it means just what I choose it to mean—neither more nor less. — Humpty Dumpty in Lewis Carroll’s Through the Looking Glass The word “graph” is commonly understood to mean a visual representation of a dataset, such as a bar chart, a histogram, a scatterplot, or a plot of a function. Examples of such graphs are shown in Figure 1.2. 1 0.5 0.5 f (x) 0 0 y −0.5 −0.5 −1 −6 −4 −2 0 2 4 6 −4 −2 0 2 4 x x (a) Plots of functions. (b) A scatterplot. Figure 1.2: Visual representations of datasets as plots. This book is not about graphs in the sense of plots of functions or datasets. Rather, our focus is on combinatorial graphs or graphs for short. A graph in the combinatorial sense is a collection of discrete interconnected elements, an example of which is shown in Figure 1.1. How can we elaborate on this brief description of combinatorial graphs? To paraphrase what Felix Klein said about curves,1 it is easy to define a graph until we realize the countless number of exceptions. There are directed graphs, weighted graphs, multigraphs, simple graphs, and so on. Where do we begin? Notation If S is a set, let S (n) denote the set of unordered n-tuples (with possible repetition). We shall sometimes refer to an unordered n-tuple as an n-set. 1 “Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.” 1.1. Graphs and digraphs 3 We start by calling a “graph” what some call an “unweighted, undirected graph without multiple edges.” Definition 1.1. A graph G = (V, E) is an ordered pair of finite sets. Elements of V are called vertices or nodes, and elements of E ⊆ V (2) are called edges or arcs. We refer to V as the vertex set of G, with E being the edge set. The cardinality of V is called the order of G, and |E| is called the size of G. We usually disregard any direction of the edges and consider (u, v) and (v, u) as one and the same edge in G. In that case, G is referred to as an undirected graph. One can label a graph by attaching labels to its vertices. If (v1 , v2 ) ∈ E is an edge of a graph G = (V, E), we say that v1 and v2 are adjacent vertices. For ease of notation, we write the edge (v1 , v2 ) as v1 v2 . The edge v1 v2 is also said to be incident with the vertices v1 and v2 . a e b d c Figure 1.3: A house graph. Example 1.2. Consider the graph in Figure 1.3. 1. List the vertex and edge sets of the graph. 2. For each vertex, list all vertices that are adjacent to it. 3. Which vertex or vertices have the largest number of adjacent vertices? Similarly, which vertex or vertices have the smallest number of adjacent vertices? 4. If all edges of the graph are removed, is the resulting figure still a graph? Why or why not? 5. If all vertices of the graph are removed, is the resulting figure still a graph? Why or why not? Solution. (1) Let G = (V, E) denote the graph in Figure 1.3. Then the vertex set of G is V = {a, b, c, d, e}. The edge set of G is given by E = {ab, ae, ba, bc, be, cb, cd, dc, de, ed, eb, ea}. (1.1) We can also use Sage to construct the graph G and list its vertex and edge sets: 4 Chapter 1. Introduction to graph theory sage : G = Graph ({ " a " :[ " b " ," e " ] , " b " :[ " a " ," c " ," e " ] , " c " :[ " b " ," d " ] , ... " d " :[ " c " ," e " ] , " e " :[ " a " ," b " ," d " ]}) sage : G Graph on 5 vertices sage : G . vertices () [ ’a ’ , ’b ’ , ’c ’ , ’d ’ , ’e ’] sage : G . edges ( labels = False ) [( ’a ’ , ’b ’) , ( ’a ’ , ’e ’) , ( ’b ’ , ’e ’) , ( ’c ’ , ’b ’) , ( ’c ’ , ’d ’) , ( ’e ’ , ’d ’ )] The graph G is undirected, meaning that we do not impose direction on any edges. Without any direction on the edges, the edge ab is the same as the edge ba. That is why G.edges() returns six edges instead of the 12 edges listed in (1.1). (2) Let adj(v) be the set of all vertices that are adjacent to v. Then we have adj(a) = {b, e} adj(b) = {a, c, e} adj(c) = {b, d} adj(d) = {c, e} adj(e) = {a, b, d}. The vertices adjacent to v are also referred to as its neighbors. We can use the function G.neighbors() to list all the neighbors of each vertex. sage : G . neighbors ( " a " ) [ ’b ’ , ’e ’] sage : G . neighbors ( " b " ) [ ’a ’ , ’c ’ , ’e ’] sage : G . neighbors ( " c " ) [ ’b ’ , ’d ’] sage : G . neighbors ( " d " ) [ ’c ’ , ’e ’] sage : G . neighbors ( " e " ) [ ’a ’ , ’b ’ , ’d ’] (3) Taking the cardinalities of the above five sets, we get |adj(a)| = |adj(c)| = |adj(d)| = 2 and |adj(b)| = |adj(e)| = 3. Thus a, c and d have the smallest number of adjacent vertices, while b and e have the largest number of adjacent vertices. (4) If all the edges in G are removed, the result is still a graph, although one without any edges. By definition, the edge set of any graph is a subset of V (2) . Removing all edges of G leaves us with the empty set ∅, which is a subset of every set. (5) Say we remove all of the vertices from the graph in Figure 1.3 and in the process all edges are removed as well. The result is that both of the vertex and edge sets are empty. This is a special graph known as the empty or null graph. Example 1.3. Consider the illustration in Figure 1.4. Does Figure 1.4 represent a graph? Why or why not? Solution. If V = {a, b, c} and E = {aa, bc}, it is clear that E ⊆ V (2) . Then (V, E) is a graph. The edge aa is called a self-loop of the graph. In general, any edge of the form vv is a self-loop. In Figure 1.3, the edges ae and ea represent one and the same edge. If we do not consider the direction of the edges in the graph of Figure 1.3, then the graph has six edges. However, if the direction of each edge is taken into account, then there are 12 edges as listed in (1.1). The following definition captures the situation where the direction of the edges are taken into account. A directed edge is an edge such that one vertex incident with it is designated as the head vertex and the other incident vertex is designated as the tail vertex. In this 1.1. Graphs and digraphs 5 a c b Figure 1.4: A figure with a self-loop. situation, we may assume that the set of edges is a subset of the ordered pairs V × V . A directed edge uv is said to be directed from its tail u to its head v. A directed graph or digraph G is a graph each of whose edges is directed. The indegree id(v) of a vertex v ∈ V (G) counts the number of edges such that v is the head of those edges. The outdegree od(v) of a vertex v ∈ V (G) is the number of edges such that v is the tail of those edges. The degree deg(v) of a vertex v of a digraph is the sum of the indegree and the outdegree of v. Let G be a graph without self-loops and multiple edges. It is important to distinguish a graph G as being directed or undirected. If G is undirected and uv ∈ E(G), then uv and vu represent the same edge. In case G is a digraph, then uv and vu are different directed edges. For a digraph G = (V, E) and a vertex v ∈ V , all the neighbors of v in G are contained in adj(v), i.e. the set of all neighbors of v. Just as we distinguish between indegree and outdegree for a vertex in a digraph, we also distinguish between in- neighbors and out-neighbors. The set of in-neighbors iadj(v) ⊆ adj(v) of v ∈ V consists of all those vertices that contribute to the indegree of v. Similarly, the set of out-neighbors oadj(v) ⊆ adj(v) of v ∈ V are those vertices that contribute to the outdegree of v. Then iadj(v) ∩ oadj(v) = {u | uv ∈ E and vu ∈ E} and adj(v) = iadj(v) ∪ oadj(v). 1.1.1 Multigraphs This subsection presents a larger class of graphs. For simplicity of presentation, in this book we shall assume usually that a graph is not a multigraph. In other words, when you read a property of graphs later in the book, it will be assumed (unless stated explicitly otherwise) that the graph is not a multigraph. However, as multigraphs and weighted graphs are very important in many applications, we will try to keep them in the back of our mind. When appropriate, we will add as a remark how an interesting property of “ordinary” graphs extends to the multigraph or weighted graph case. An important class of graphs consist of those graphs having multiple edges between pairs of vertices. A multigraph is a graph in which there are multiple edges between a pair of vertices. A multi-undirected graph is a multigraph that is undirected. Similarly, a multidigraph is a directed multigraph. Example 1.4. Sage can compute with and plot multigraphs, or multidigraphs, having loops. sage : G = Graph ({0:{0: ’ e0 ’ ,1: ’ e1 ’ ,2: ’ e2 ’ ,3: ’ e3 ’} , 2:{5: ’ e4 ’ }}) sage : G . show ( vertex_labels = True , edge_labels = True , graph_border = True ) sage : H = DiGraph ({0:{0: " e0 " }} , Loops = True ) sage : H . add_edges ([(0 ,1 , ’ e1 ’) , (0 ,2 , ’ e2 ’) , (0 ,2 , ’ e3 ’) , (1 ,2 , ’ e4 ’) , (1 ,0 , ’ e5 ’ )]) sage : H . show ( vertex_labels = True , edge_labels = True , graph_border = True ) 6 Chapter 1. Introduction to graph theory 5 e4 2 2 e3 e4 e2 e2 e5 0 e0 1 0 e3 e1 3 e0 1 e1 (a) (b) Figure 1.5: A graph G and digraph H with a loop and multi-edges. These graphs are plotted in Figure 1.5. As we indicated above, a graph may have “weighted” edges. Definition 1.5. A weighted graph is a graph G = (V, E) where each set V and E is a pair consisting of a vertex and a real number called the weight. The illustration in Figure 1.1 is actually a multigraph, a graph with multiple edges, called the Königsberg graph. Definition 1.6. For a weighted multigraph G, we are given: A finite set V whose elements are pairs (v, wv ), where v is called a vertex and wv ∈ R is the vertex weight. (Sometimes, the pair (v, wv ) is called the vertex.) A finite set E whose elements are weighted edges. We do not necessarily assume that E ⊆ V (2) , where V (2) is the set of unordered pairs of vertices.2 Each weighted edge can be represented as a 3-tuple of the form (we , u, v), where (u, v) is the edge in question and we ∈ R is the edge weight. An incidence function i : E → V (2) . (1.2) Such a multigraph is denoted G = (V, E, i). An orientation on G is a function h:E→V (1.3) where h(e) ∈ i(e) for all e ∈ E. The element v = h(e) is called the head of i(e). If G has no self-loops, then i(e) is a set having exactly two elements denoted i(e) = {h(e), t(e)}. The element v = t(e) is called the tail of i(e). For self-loops, we set t(e) = h(e). A multigraph with an orientation can therefore be described as the 4-tuple (V, E, i, h). In other words, G = (V, E, i, h) is a multidigraph. Figure 1.6 illustrates a weighted multigraph. 2 However, we always assume that E ⊆ R × V (2) , where the R-component is called the weight of the edge. 1.1. Graphs and digraphs 7 1 v3 v4 2 3 1 1 1 2 3 v5 3 3 6 v2 v1 1 Figure 1.6: An example of a weighted multigraph. The vertex degree of a weighted multigraph must be defined. There is a weighted degree and an unweighted degree. Let G be a graph as in Definition 1.6. The unweighted indegree of a vertex v ∈ V counts the edges going into v: X deg+ (v) = 1. e∈E h(e)=v The unweighted outdegree of a vertex v ∈ V counts the edges going out of v: X deg− (v) = 1. e∈E v∈i(e)={v,v 0 } h(e)=v 0 The unweighted degree deg(v) of a vertex v of a weighted multigraph is the sum of the unweighted indegree and the unweighted outdegree of v: deg(v) = deg+ (v) + deg− (v). (1.4) Loops are counted twice. Similarly, there is the set of in-neighbors iadj(v) = {w ∈ V | for some e ∈ E, i(e) = {v, w}, h(e) = v} and the set of out-neighbors oadj(v) = {w ∈ V | for some e ∈ E, i(e) = {v, w}, h(e) = w}. Define the adjacency of v to be the union of these: adj(v) = iadj(v) ∪ oadj(v). (1.5) It is clear that deg+ (v) = | iadj(v)| and deg− (v) = | oadj(v)|. The weighted indegree of a vertex v ∈ V counts the weights of edges going into v: X wdeg + (v) = wv . e∈E h(e)=v 8 Chapter 1. Introduction to graph theory The weighted outdegree of a vertex v ∈ V counts the weights of edges going out of v: X wdeg − (v) = wv . e∈E v∈i(e)={v,v 0 } h(e)=v 0 The weighted degree of a vertex of a weighted multigraph is the sum of the weighted indegree and the weighted outdegree of that vertex, wdeg(v) = wdeg + (v) + wdeg − (v). In other words, it is the sum of the weights of the edges incident to that vertex, regarding the graph as an undirected weighted graph. Unweighted degrees are a special case of weighted degrees. For unweighted degrees, we merely set each edge weight to unity. Definition 1.7. Let G = (V, E, h) be an unweighted multidigraph. The line graph of G, denoted L(G), is the multidigraph whose vertices are the edges of G and whose edges are (e, e0 ) where h(e) = t(e0 ) (for e, e0 ∈ E). A similar definition holds if G is undirected. For example, the line graph of the cyclic graph is itself. 1.1.2 Simple graphs Our life is frittered away by detail. . . . Simplify, simplify. Instead of three meals a day, if it be necessary eat but one; instead of a hundred dishes, five; and reduce other things in proportion. — Henry David Thoreau, Walden, 1854, Chapter 2: Where I Lived, and What I Lived For A simple graph is a graph with no self-loops and no multiple edges. Figure 1.7 illustrates a simple graph and its digraph version, together with a multidigraph version of the Königsberg graph. The edges of a digraph can be visually represented as directed arrows, similar to the digraph in Figure 1.7(b) and the multidigraph in Figure 1.7(c). The digraph in Figure 1.7(b) has the vertex set {a, b, c} and the edge set {ab, bc, ca}. There is an arrow from vertex a to vertex b, hence ab is in the edge set. However, there is no arrow from b to a, so ba is not in the edge set of the graph in Figure 1.7(b). The family Sh(n) of Shannon multigraphs is illustrated in Figure 1.8 for integers 2 ≤ n ≤ 7. These graphs are named after Claude E. Shannon (1916–2001) and are sometimes used when studying edge colorings. Each Shannon multigraph consists of three vertices, giving rise to a total of three distinct unordered pairs. Two of these pairs are connected by bn/2c edges and the third pair of vertices is connected by b(n + 1)/2c edges. Notational convention Unless stated otherwise, all graphs are simple graphs in the remainder of this book. Definition 1.8. For any vertex v in a graph G = (V, E), the cardinality of adj(v) (as in 1.5) is called the degree of v and written as deg(v) = | adj(v)|. The degree of v counts the number of vertices in G that are adjacent to v. If deg(v) = 0, then v is not incident to any edge and we say that v is an isolated vertex. If G has no loops and deg(v) = 1, then v is called a pendant. 1.1. Graphs and digraphs 9 c a a b d c b c b a (a) Simple graph. (b) Digraph. (c) Multidigraph. Figure 1.7: A simple graph, its digraph version, and a multidigraph. (a) Sh(2) (b) Sh(3) (c) Sh(4) (d) Sh(5) (e) Sh(6) (f) Sh(7) Figure 1.8: The family of Shannon multigraphs Sh(n) for n = 2, . . . , 7. 10 Chapter 1. Introduction to graph theory Some examples would put the above definition in concrete terms. Consider again the graph in Figure 1.4. Note that no vertices are isolated. Even though vertex a is not incident to any vertex other than a itself, note that deg(a) = 2 and so by definition a is not isolated. Furthermore, each of b and c is a pendant. For the house graph in Figure 1.3, we have deg(b) = 3. For the graph in Figure 1.7(b), we have deg(b) = 2. If V 6= ∅ and E = ∅, then G is a graph consisting entirely of isolated vertices. From Example 1.2 we know that the vertices a, c, d in Figure 1.3 have the smallest degree in the graph of that figure, while b, e have the largest degree. The minimum degree among all vertices in G is denoted δ(G), whereas the maximum degree is written as ∆(G). Thus, if G denotes the graph in Figure 1.3 then we have δ(G) = 2 and ∆(G) = 3. In the following Sage session, we construct the digraph in Figure 1.7(b) and compute its maximum and minimum number of degrees. sage : G = DiGraph ({ " a " : " b " , " b " : " c " , " c " : " a " }) sage : G Digraph on 3 vertices sage : G . degree ( " a " ) 2 sage : G . degree ( " b " ) 2 sage : G . degree ( " c " ) 2 So for the graph G in Figure 1.7, we have δ(G) = ∆(G) = 2. The graph G in Figure 1.7 has the special property that its minimum degree is the same as its maximum degree, i.e. δ(G) = ∆(G). Graphs with this property are referred to as regular . An r-regular graph is a regular graph each of whose vertices has degree r. For instance, G is a 2-regular graph. The following result, due to Euler, counts the total number of degrees in any graph. P Theorem 1.9. Euler 1736. If G = (V, E) is a graph, then v∈V deg(v) = 2|E|. Proof. Each edge e = v1 v2 ∈ E is incident with two vertices, so e is counted twice towards the total sum of degrees. The first time, we count e towards the degree of vertex v1 and the second time we count e towards the degree of v2 . Theorem 1.9 is sometimes called the “handshaking lemma,” due to its interpretation as in the following story. Suppose you go into a room. Suppose there are n people in the room (including yourself) and some people shake hands with others and some do not. Create the graph with n vertices, where each vertex is associated with a different person. Draw an edge between two people if they shook hands. The degree of a vertex is the number of times that person has shaken hands (we assume that there are no multiple edges, i.e. that no two people shake hands twice). The theorem above simply says that the total number of handshakes is even. This is “obvious” when you look at it this way since each handshake is counted twice (A shaking B’s hand is counted, and B shaking A’s hand is counted as well, since the sum in the theorem is over all vertices). To interpret Theorem 1.9 in a slightly different way within the context of the same room of people, there is an even number of people who shook hands with an odd number of other people. This consequence of Theorem 1.9 is recorded in the following corollary. Corollary 1.10. A graph G = (V, E) contains an even number of vertices with odd degrees. Proof. Partition V into two disjoint subsets: Ve is the subset of V that contains only vertices with even degrees; and Vo is the subset of V with only vertices of odd degrees. 1.2. Subgraphs and other graph types 11 That is, V = Ve ∪ Vo and Ve ∩ Vo = ∅. From Theorem 1.9, we have X X X deg(v) = deg(v) + deg(v) = 2|E| v∈V v∈Ve v∈Vo which can be re-arranged as X X X deg(v) = deg(v) − deg(v). v∈Vo v∈V v∈Ve P P As v∈V deg(v) and v∈Ve deg(v) are both even, their difference is also even. As E ⊆ V (2) , then E can be the empty set, in which case the total degree of G = (V, E) is zero. Where E 6= ∅, then the total degree of G is greater than zero. By Theorem 1.9, the total degree of G is nonnegative and even. This result is an immediate consequence of Theorem 1.9 and is captured in the following corollary. Corollary 1.11. If G is a graph, then the sum of its vertex degrees is nonnegative and even. If G = (V, E) is an r-regular graph with n vertices and m edges, it is clear by definition of r-regular graphs that the total degree of G is rn. By Theorem 1.9 we have 2m = rn and therefore m = rn/2. This result is captured in the following corollary. Corollary 1.12. If G = (V, E) is an r-regular graph having n vertices and m edges, then m = rn/2. 1.2 Subgraphs and other graph types We now consider several common types of graphs. Along the way, we also present basic properties of graphs that could be used to distinguish different types of graphs. Let G be a multigraph as in Definition 1.6, with vertex set V (G) and edge set E(G). Consider a graph H such that V (H) ⊆ V (G) and E(H) ⊆ E(G). Furthermore, if e ∈ E(H) and i(e) = {u, v}, then u, v ∈ V (H). Under these conditions, H is called a subgraph of G. 1.2.1 Walks, trails, and paths I like long walks, especially when they are taken by people who annoy me. — Noel Coward If u and v are two vertices in a graph G, a u-v walk is an alternating sequence of vertices and edges starting with u and ending at v. Consecutive vertices and edges are incident. Formally, a walk W of length n ≥ 0 can be defined as W : v0 , e1 , v1 , e2 , v2 , . . . , vn−1 , en , vn where each edge ei = vi−1 vi and the length n refers to the number of (not necessarily distinct) edges in the walk. The vertex v0 is the starting vertex of the walk and vn is the end vertex, so we refer to W as a v0 -vn walk. The trivial walk is the walk of length n = 0 in which the start and end vertices are one and the same vertex. If the graph has 12 Chapter 1. Introduction to graph theory no multiple edges then, for brevity, we omit the edges in a walk and usually write the walk as the following sequence of vertices: W : v0 , v1 , v2 , . . . , vn−1 , vn . For the graph in Figure 1.9, an example of a walk is an a-e walk: a, b, c, b, e. In other words, we start at vertex a and travel to vertex b. From b, we go to c and then back to b again. Then we end our journey at e. Notice that consecutive vertices in a walk are adjacent to each other. One can think of vertices as destinations and edges as footpaths, say. We are allowed to have repeated vertices and edges in a walk. The number of edges in a walk is called its length. For instance, the walk a, b, c, b, e has length 4. a b c g d f e Figure 1.9: Walking along a graph. A trail is a walk with no repeating edges. For example, the a-b walk a, b, c, d, f, g, b in Figure 1.9 is a trail. It does not contain any repeated edges, but it contains one repeated vertex, i.e. b. Nothing in the definition of a trail restricts a trail from having repeated vertices. A walk with no repeating vertices, except possibly the first and last, is called a path. Without any repeating vertices, a path cannot have repeating edges, hence a path is also a trail. Proposition 1.13. Let G = (V, E) be a simple (di)graph of order n = |V |. Any path in G has length at most n − 1. Proof. Let V = {v1 , v2 , . . . , vn } be the vertex set of G. Without loss of generality, we can assume that each pair of vertices in the digraph G is connected by an edge, giving a total of n2 possible edges for E = V × V . We can remove self-loops from E, which now leaves us with an edge set E1 that consists of n2 − n edges. Start our path from any vertex, say, v1 . To construct a path of length 1, choose an edge v1 vj1 ∈ E1 such that vj1 ∈ / {v1 }. Remove from E1 all v1 vk such that vj1 6= vk . This results in a reduced edge set E2 of n2 − n − (n − 2) elements and we now have the path P1 : v1 , vj1 of length 1. Repeat the same process for vj1 vj2 ∈ E2 to obtain a reduced edge set E3 of n2 − n − 2(n − 2) elements and a path P2 : v1 , vj1 , vj2 of length 2. In general, let Pr : v1 , vj1 , vj2 , . . . , vjr be a path of length r < n and let Er+1 be our reduced edge set of n2 − n − r(n − 2) elements. Repeat the above process until we have constructed a path Pn−1 : v1 , vj1 , vj2 , . . . , vjn−1 of length n − 1 with reduced edge set En of n2 − n − (n − 1)(n − 2) elements. Adding another vertex to Pn−1 means going back to a vertex that was previously visited, because Pn−1 already contains all vertices of V . 1.2. Subgraphs and other graph types 13 A walk of length n ≥ 3 whose start and end vertices are the same is called a closed walk . A trail of length n ≥ 3 whose start and end vertices are the same is called a closed trail . A path of length n ≥ 3 whose start and end vertices are the same is called a closed path or a cycle (with apologies for slightly abusing terminology).3 For example, the walk a, b, c, e, a in Figure 1.9 is a closed path. A path whose length is odd is called odd , otherwise it is referred to as even. Thus the walk a, b, e, a in Figure 1.9 is a cycle. It is easy to see that if you remove any edge from a cycle, then the resulting walk contains no closed walks. An Euler subgraph of a graph G is either a cycle or an edge-disjoint union of cycles in G. An example of a closed walk which is not a cycle is given in Figure 1.10. 0 4 2 1 3 Figure 1.10: Butterfly graph with 5 vertices. The length of the shortest cycle in a graph is called the girth of the graph. By convention, an acyclic graph is said to have infinite girth. Example 1.14. Consider the graph in Figure 1.9. 1. Find two distinct walks that are not trails and determine their lengths. 2. Find two distinct trails that are not paths and determine their lengths. 3. Find two distinct paths and determine their lengths. 4. Find a closed trail that is not a cycle. 5. Find a closed walk C which has an edge e such that C − e contains a cycle. Solution. (1) Here are two distinct walks that are not trails: w1 : g, b, e, a, b, e and w2 : f, d, c, e, f, d. The length of walk w1 is 5 and the length of walk w2 is also 5. (2) Here are two distinct trails that are not paths: t1 : a, b, c, e, b and t2 : b, e, f, d, c, e. The length of trail t1 is 4 and the length of trail t2 is 5. (3) Here are two distinct paths: p1 : a, b, c, d, f, e and p2 : g, b, a, e, f, d. The length of path p1 is 5 and the length of path p2 is also 5. (4) Here is a closed trail that is not a cycle: d, c, e, b, a, e, f, d. (5) Left as an exercise. Theorem 1.15. Every u-v walk in a graph contains a u-v path. Proof. A walk of length n = 0 is the trivial path. So assume that W is a u-v walk of length n > 0 in a graph G: W : u = v0 , v1 , . . . , vn = v. It is possible that a vertex in W is assigned two different labels. If W has no repeated vertices, then W is already a path. Otherwise W has at least one repeated vertex. Let 3 A cycle in a graph is sometimes also called a “circuit”. Since that terminology unfortunately conflicts with the closely related notion of a circuit of a matroid, we do not use it here. 14 Chapter 1. Introduction to graph theory 0 ≤ i, j ≤ n be two distinct integers with i < j such that vi = vj . Deleting the vertices vi , vi+1 , . . . , vj−1 from W results in a u-v walk W1 whose length is less than n. If W1 is a path, then we are done. Otherwise we repeat the above process to obtain a u-v walk shorter than W1 . As W is a finite sequence, we only need to apply the above process a finite number of times to arrive at a u-v path. A graph is said to be connected if for every pair of distinct vertices u, v there is a u-v path joining them. A graph that is not connected is referred to as disconnected . The empty graph is disconnected and so is any nonempty graph with an isolated vertex. However, the graph in Figure 1.7 is connected. A geodesic path or shortest path between two distinct vertices u, v of a graph is a u-v path of minimum length. A nonempty graph may have several shortest paths between some distinct pair of vertices. For the graph in Figure 1.9, both a, b, c and a, e, c are geodesic paths between a and c. Let H be a connected subgraph of a graph G such that H is not a proper subgraph of any connected subgraph of G. Then H is said to be a component of G. We also say that H is a maximal connected subgraph of G. Any connected graph is its own component. The number of connected components of a graph G will be denoted ω(G). The following is an immediate consequence of Corollary 1.10. Proposition 1.16. Suppose that exactly two vertices of a graph have odd degree. Then those two vertices are connected by a path. Proof. Let G be a graph all of whose vertices are of even degree, except for u and v. Let C be a component of G containing u. By Corollary 1.10, C also contains v, the only remaining vertex of odd degree. As u and v belong to the same component, they are connected by a path. Example 1.17. Determine whether or not the graph in Figure 1.9 is connected. Find a shortest path from g to d. Solution. In the following Sage session, we first construct the graph in Figure 1.9 and use the method is_connected() to determine whether or not the graph is connected. Finally, we use the method shortest_path() to find a geodesic path between g and d. sage : g = Graph ({ " a " :[ " b " ," e " ] , " b " :[ " a " ," g " ," e " ," c " ] , \ ... " c " :[ " b " ," e " ," d " ] , " d " :[ " c " ," f " ] , " e " :[ " f " ," a " ," b " ," c " ] , \ ... " f " :[ " g " ," d " ," e " ] , " g " :[ " b " ," f " ]}) sage : g . is_connected () True sage : g . shortest_path ( " g " , " d " ) [ ’g ’ , ’f ’ , ’d ’] This shows that g, f, d is a shortest path from g to d. In fact, any other g-d path has length greater than 2, so we can say that g, f, d is the shortest path between g and d. Remark 1.18. We will explain Dijkstra’s algorithm in Chapter 3. Dijkstra’s algorithm gives one of the best algorithms for finding shortest paths between two vertices in a connected graph. What is very remarkable is that, at the present state of knowledge, finding the shortest path from a vertex v to a particular (but arbitrarily given) vertex w appears to be as hard as finding the shortest path from a vertex v to all other vertices in the graph! Trees are a special type of graphs that are used in modelling structures that have some form of hierarchy. For example, the hierarchy within an organization can be drawn as a tree structure, similar to the family tree in Figure 1.11. Formally, a tree is an 1.2. Subgraphs and other graph types 15 undirected graph that is connected and has no cycles. If one vertex of a tree is specially designated as the root vertex , then the tree is called a rooted tree. Chapter 2 covers trees in more details. grandma mum uncle aunt me sister brother cousin1 cousin2 Figure 1.11: A family tree. 1.2.2 Subgraphs, complete and bipartite graphs Let G be a graph with vertex set V (G) and edge set E(G). Suppose we have a graph H such that V (H) ⊆ V (G) and E(H) ⊆ E(G). Furthermore, suppose the incidence function i of G, when restricted to E(H), has image in V (H)(2) . Then H is a subgraph of G. In this situation, G is referred to as a supergraph of H. Starting from G, one can obtain its subgraph H by deleting edges and/or vertices from G. Note that when a vertex v is removed from G, then all edges incident with v are also removed. If V (H) = V (G), then H is called a spanning subgraph of G. In Figure 1.12, let G be the left-hand side graph and let H be the right-hand side graph. Then it is clear that H is a spanning subgraph of G. To obtain a spanning subgraph from a given graph, we delete edges from the given graph. (a) (b) Figure 1.12: A graph and one of its subgraphs. We now consider several standard classes of graphs. The complete graph Kn on n vertices is a graph such that any two distinct vertices are adjacent. As |V (Kn )| = n, then |E(Kn )| is equivalent to the total number of 2-combinations from a set of n objects: n n(n − 1) |E(Kn )| = = . (1.6) 2 2 Thus for any simple graph G with n vertices, its total number of edges |E(G)| is bounded above by n(n − 1) |E(G)| ≤ . (1.7) 2 16 Chapter 1. Introduction to graph theory Figure 1.13 shows complete graphs each of whose total number of vertices is bounded by 1 ≤ n ≤ 5. The complete graph K1 has one vertex with no edges. It is also called the trivial graph. (a) K5 (b) K4 (c) K3 (d) K2 (e) K1 Figure 1.13: Complete graphs Kn for 1 ≤ n ≤ 5. The following result is an application of inequality (1.7). Theorem 1.19. Let G be a simple graph with n vertices and k components. Then G has at most 21 (n − k)(n − k + 1) edges. Proof. If ni is the number of vertices in component i, then ni > 0 and it can be shown (see the proof of Lemma 2.1 in [80, pp.21–22]) that X X 2 X n2i ≤ ni − (k − 1) 2 ni − k . (1.8) (This P result holds true for any nonempty but finite set of positive integers.) Note that ni = n and by (1.7) each component i has at most 21 ni (ni − 1) edges. Apply (1.8) to get X ni (ni − 1) 1X 2 1X = ni − ni 2 2 2 1 1 ≤ (n2 − 2nk + k 2 + 2n − k) − n 2 2 (n − k)(n − k + 1) = 2 as required. The cycle graph on n ≥ 3 vertices, denoted Cn , is the connected 2-regular graph on n vertices. Each vertex in Cn has degree exactly 2 and Cn is connected. Figure 1.14 shows cycles graphs Cn where 3 ≤ n ≤ 6. The path graph on n ≥ 1 vertices is denoted Pn . For n = 1, 2 we have P1 = K1 and P2 = K2 . Where n ≥ 3, then Pn is a spanning subgraph of Cn obtained by deleting one edge. A bipartite graph G is a graph with at least two vertices such that V (G) can be split into two disjoint subsets V1 and V2 , both nonempty. Every edge uv ∈ E(G) is such that u ∈ V1 and v ∈ V2 , or v ∈ V1 and u ∈ V2 . See Kalman [116] for an application of bipartite graphs to the problem of allocating satellites to radio stations. Example 1.20. The Franklin graph, shown in Figure 1.15, is named after Philip Franklin. It is a 3-regular graph with 12 vertices and 18 edges. It is bipartite, Hamiltonian and has radius 3, diameter 3 and girth 4. It is also a 3-vertex-connected and 3-edge-connected perfect graph. 1.2. Subgraphs and other graph types 17 (a) C6 (b) C5 (c) C4 (d) C3 Figure 1.14: Cycle graphs Cn for 3 ≤ n ≤ 6. sage : G = graphs . LCFGraph (12 , [5 , -5] , 6) sage : G . show ( dpi =300) sage : G . is_bipartite () True sage : G . chromatic_number () 2 sage : G . girth () 4 sage : G . is_hamiltonian () True sage : G . i s_vertex_transitive () True sage : G . is_planar () False sage : G . is_regular () True sage : G . coloring () [[1 , 3 , 5 , 7 , 9 , 11] , [0 , 2 , 4 , 6 , 8 , 10]] Figure 1.15: Franklin graph example. Example 1.21. The Foster graph, shown in Figure 1.16, is a 3-regular graph with 90 vertices and 135 edges. This is a bipartite, Hamiltonian graph that has radius 8, diameter 18 Chapter 1. Introduction to graph theory 8 and girth 10. It is also a 3-vertex-connected and 3-edge-connected graph. sage : G = graphs . LCFGraph (90 , [17 , -9 ,37 , -37 ,9 , -17] , 15) sage : G . plot ( vertex_labels = False , vertex_size =0 , graph_border = True ). show ( dpi =300) sage : G . i s_vertex_transitive () True sage : G . is_hamiltonian () True sage : G . girth () 10 sage : G . is_bipartite () True sage : len ( G . vertices ()) 90 sage : len ( G . edges ()) 135 Figure 1.16: Foster graph example. Theorem 1.22. A graph is bipartite if and only if it has no odd cycles. Proof. Necessity (=⇒): Assume G to be bipartite. Traversing each edge involves going from one side of the bipartition to the other. For a walk to be closed, it must have 1.3. Representing graphs in a computer 19 even length in order to return to the side of the bipartition from which the walk started. Thus, any cycle in G must have even length. Sufficiency (⇐=): Assume G = (V, E) has order n ≥ 2 and no odd cycles. If G is connected, choose any vertex u ∈ V and define a partition of V thus: X = {x ∈ V | d(u, x) is even}, Y = {y ∈ V | d(u, y) is odd} where d(u, v) denotes the distance (or length of the shortest path) from u to v. If (X, Y ) is a bipartition of G, then we are done. Otherwise, (X, Y ) is not a bipartition of G. Then one of X and Y has two vertices v, w joined by an edge e. Let P1 be a shortest u-v path and P2 be a shortest u-w path. By definition of X and Y , both P1 and P2 have even lengths or both have odd lengths. From u, let x be the last vertex common to both P1 and P2 . The subpath u-x of P1 and u-x of P2 have equal length. That is, the subpath x-v of P1 and x-w of P2 both have even or odd lengths. Construct a cycle C from the paths x-v and x-w, and the edge e joining v and w. Since x-v and x-w both have even or odd lengths, the cycle C has odd length, contradicting our hypothesis that G has no odd cycles. Hence, (X, Y ) is a bipartition of G. Finally, if G is disconnected, each of its components has no odd cycles. Repeat the above argument for each component to conclude that G is bipartite. Example 1.23. The Gray graph, shown in Figure 1.17, is an undirected bipartite graph with 54 vertices and 81 edges. It is a 3-regular graph discovered by Marion C. Gray in 1932. The Gray graph has chromatic number 2, chromatic index 3, radius 6, and diameter 6. It is also a 3-vertex-connected and 3-edge-connected non-planar graph. The Gray graph is an example of a graph which is edge-transitive but not vertex-transitive. sage : G = graphs . LCFGraph (54 , [ -25 ,7 , -7 ,13 , -13 ,25] , 9) sage : G . plot ( vertex_labels = False , vertex_size =0 , graph_border = True ) sage : G . is_bipartite () True sage : G . i s_vertex_transitive () False sage : G . is_hamiltonian () True sage : G . diameter () 6 The complete bipartite graph Km,n is the bipartite graph whose vertex set is parti- tioned into two nonempty disjoint sets V1 and V2 with |V1 | = m and |V2 | = n. Any vertex in V1 is adjacent to each vertex in V2 , and any two distinct vertices in Vi are not adjacent to each other. If m = n, then Kn,n is n-regular. Where m = 1 then K1,n is called the star graph. Figure 1.18 shows a bipartite graph together with the complete bipartite graphs K4,3 and K3,3 , and the star graph K1,4 . As an example of K3,3 , suppose that there are 3 boys and 3 girls dancing in a room. The boys and girls naturally partition the set of all people in the room. Construct a graph having 6 vertices, each vertex corresponding to a person in the room, and draw an edge form one vertex to another if the two people dance together. If each girl dances three times, once with with each of the three boys, then the resulting graph is K3,3 . 1.3 Representing graphs in a computer Neo: What is the Matrix? Morpheus: Unfortunately, no one can be told what the Matrix is. You have to see it for 20 Chapter 1. Introduction to graph theory Figure 1.17: Gray graph example. (a) bipartite (b) K4,3 (c) K3,3 (d) K1,4 Figure 1.18: Bipartite, complete bipartite, and star graphs. 1.3. Representing graphs in a computer 21 yourself. — From the movie The Matrix, 1999 An m × n matrix A can be represented as a11 a12 · · · a1n a21 a22 · · · a2n A= . . . . . . . . . . . . . . . . . . . . am1 am2 · · · amn The positive integers m and n are the row and column dimensions of A, respectively. The entry in row i column j is denoted aij . Where the dimensions of A are clear from context, A is also written as A = [aij ]. Representing a graph as a matrix is very inefficient in some cases and not so in other cases. Imagine you walk into a large room full of people and you consider the “handshaking graph” discussed in connection with Theorem 1.9. If not many people shake hands in the room, it is a waste of time recording all the handshakes and also all the “non-handshakes.” This is basically what the adjacency matrix does. In this kind of “sparse graph” situation, it would be much easier to simply record the handshakes as a Python dictionary.4 This section requires some concepts and techniques from linear algebra, especially matrix theory. See introductory texts on linear algebra and matrix theory [19] for coverage of such concepts and techniques. 1.3.1 Adjacency matrix Let G be an undirected graph with vertices V = {v1 , . . . , vn } and edge set E. The adjacency matrix of G is the n × n matrix A = [aij ] defined by ( 1, if vi vj ∈ E, aij = 0, otherwise. The adjacency matrix of G is also written as A(G). As G is an undirected graph, then A is a symmetric matrix. That is, A is a square matrix such that aij = aji . Now let G be a directed graph with vertices V = {v1 , . . . , vn } and edge set E. The (0, −1, 1)-adjacency matrix of G is the n × n matrix A = [aij ] defined by 1, if vi vj ∈ E, aij = −1, if vj vi ∈ E, 0, otherwise. Example 1.24. Compute the adjacency matrices of the graphs in Figure 1.19. Solution. Define the graphs in Figure 1.19 using DiGraph and Graph. Then call the method adjacency_matrix(). 4 A Python dictionary is basically an indexed set. See the reference manual at http://www.python.org for further details. 22 Chapter 1. Introduction to graph theory 3 f 6 2 d e 5 1 b c 4 a (a) (b) Figure 1.19: What are the adjacency matrices of these graphs? sage : G1 = DiGraph ({1:[2] , 2:[1] , 3:[2 ,6] , 4:[1 ,5] , 5:[6] , 6:[5]}) sage : G2 = Graph ({ " a " :[ " b " ," c " ] , " b " :[ " a " ," d " ] , " c " :[ " a " ," e " ] , \ ... " d " :[ " b " ," f " ] , " e " :[ " c " ," f " ] , " f " :[ " d " ," e " ]}) sage : m1 = G1 . adjacency_matrix (); m1 [0 1 0 0 0 0] [1 0 0 0 0 0] [0 1 0 0 0 1] [1 0 0 0 1 0] [0 0 0 0 0 1] [0 0 0 0 1 0] sage : m2 = G2 . adjacency_matrix (); m2 [0 1 1 0 0 0] [1 0 0 1 0 0] [1 0 0 0 1 0] [0 1 0 0 0 1] [0 0 1 0 0 1] [0 0 0 1 1 0] sage : m1 . is_symmetric () False sage : m2 . is_symmetric () True In general, the adjacency matrix of a digraph is not symmetric, while that of an undi- rected graph is symmetric. More generally, if G is an undirected multigraph with edge eij = vi vj having mul- tiplicity wij , or a weighted graph with edge eij = vi vj having weight wij , then we can define the (weighted) adjacency matrix A = [aij ] by ( wij , if vi vj ∈ E, aij = 0, otherwise. For example, Sage allows you to easily compute a weighted adjacency matrix. sage : G = Graph ( sparse = True , weighted = True ) sage : G . add_edges ([(0 ,1 ,1) , (1 ,2 ,2) , (0 ,2 ,3) , (0 ,3 ,4)]) sage : M = G . w e i g h t e d _ a d ja c e n c y _ m a t r i x (); M [0 1 3 4] [1 0 2 0] [3 2 0 0] [4 0 0 0] Bipartite case Suppose G = (V, E) is an undirected bipartite graph with n = |V | vertices. Any ad- jacency matrix A of G is symmetric and we assume that it is indexed from zero up to 1.3. Representing graphs in a computer 23 n − 1, inclusive. Then there exists a permutation π of the index set {0, 1, . . . , n − 1} such that the matrix A0 = [aπ(i)π(j) ] is also an adjacency matrix for G and has the form 0 0 B A = (1.9) BT 0 where 0 is a zero matrix. The matrix B is called a reduced adjacency matrix or a bi- adjacency matrix (the literature also uses the terms “transfer matrix” or the ambiguous term “adjacency matrix”). In fact, it is known [9, p.16] that any undirected graph is bipartite if and only if there is a permutation π on {0, 1, . . . , n − 1} such that A0 (G) = [aπ(i)π(j) ] can be written as in (1.9). Tanner graphs If H is an m × n (0, 1)-matrix, then the Tanner graph of H is the bipartite graph G = (V, E) whose set of vertices V = V1 ∪V2 is partitioned into two sets: V1 corresponding to the m rows of H and V2 corresponding to the n columns of H. For any i, j with 1 ≤ i ≤ m and 1 ≤ j ≤ n, there is an edge ij ∈ E if and only if the (i, j)-th entry of H is 1. This matrix H is sometimes called the reduced adjacency matrix or the check matrix of the Tanner graph. Tanner graphs are used in the theory of error-correcting codes. For example, Sage allows you to easily compute such a bipartite graph from its matrix. sage : H = Matrix ([(1 ,1 ,1 ,0 ,0) , (0 ,0 ,1 ,0 ,1) , (1 ,0 ,0 ,1 ,1)]) sage : B = BipartiteGraph ( H ) sage : B . r ed u c e d_ a d j ac e n c y_ m a t ri x () [1 1 1 0 0] [0 0 1 0 1] [1 0 0 1 1] sage : B . plot ( graph_border = True ) The corresponding graph is similar to that in Figure 1.20. 1 1 2 3 2 4 5 3 Figure 1.20: A Tanner graph. Theorem 1.25. Let A be the adjacency matrix of a graph G with vertex set V = {v1 , v2 , . . . , vp }. For each positive integer n, the ij-th entry of An counts the number of vi -vj walks of length n in G. Proof. We shall prove by induction on n. For the base case n = 1, the ij-th entry of A1 counts the number of walks of length 1 from vi to vj . This is obvious because A1 is merely the adjacency matrix A. Suppose for induction that for some positive integer k ≥ 1, the ij-th entry of Ak counts the number of walks of length k from vi to vj . We need to show that the ij-th 24 Chapter 1. Introduction to graph theory entry of Ak+1 counts the number of vi -vj walks of length k + 1. Let A = [aij ], Ak = [bij ], and Ak+1 = [cij ]. Since Ak+1 = AAk , then p X cij = air brj r=1 for i, j = 1, 2, . . . , p. Note that air is the number of edges from vi to vr , and brj is the number of vr -vj walks of length k. Any edge from vi to vr can be joined with any vr -vj walk to create a walk vi , vr , . . . , vj of length k + 1. Then for each r = 1, 2, . . . , p, the value air brj counts the number of vi -vj walks of length k + 1 with vr being the second vertex in the walk. Thus cij counts the total number of vi -vj walks of length k + 1. 1.3.2 Incidence matrix The relationship between edges and vertices provides a very strong constraint on the data structure, much like the relationship between points and blocks in a combinatorial design or points and lines in a finite plane geometry. This incidence structure gives rise to another way to describe a graph using a matrix. Let G be a digraph with edge set E = {e1 , . . . , em } and vertex set V = {v1 , . . . , vn }. The incidence matrix of G is the n × m matrix B = [bij ] defined by −1, if vi is the tail of ej , 1, if vi is the head of ej , bij = (1.10) 2, if ej is a self-loop at vi , 0, otherwise. Each column of B corresponds to an edge and each row corresponds to a vertex. The definition of incidence matrix of a digraph as contained in expression (1.10) is applicable to digraphs with self-loops as well as multidigraphs. For the undirected case, let G be an undirected graph with edge set E = {e1 , . . . , em } and vertex set V = {v1 , . . . , vn }. The unoriented incidence matrix of G is the n × m matrix B = [bij ] defined by 1, if vi is incident to ej , bij = 2, if ej is a self-loop at vi , 0, otherwise. An orientation of an undirected graph G is an assignment of direction to each edge of G. In other words, each edge has a distinguished vertex called a head. In this case, the letter D = D(G) is sometimes used instead of B for the incidence matrix of a digraph or an oriented graph. The oriented incidence matrix D of G is defined similarly to the case where G is a digraph: it is the incidence matrix of any orientation of G. For each column of D, we have 1 as an entry in the row corresponding to one vertex of the edge under consideration and −1 as an entry in the row corresponding to the other vertex. Similarly, dij = 2 if ej is a self-loop at vi . Sage allows you to compute the incidence matrix of a graph: 1.3. Representing graphs in a computer 25 sage : G = Graph ({1: [2 , 4] , 2: [1 , 3] , 3: [2 , 6] , 4: [1 , 5] , 5: [4 , 6] , 6: [3 , 5]}) sage : G . incidence_matrix () [ -1 -1 0 0 0 0] [ 0 1 -1 0 0 0] [ 0 0 1 -1 0 0] [ 1 0 0 0 -1 0] [ 0 0 0 0 1 -1] [ 0 0 0 1 0 1] The integral cycle space of a graph is equal to the kernel of an oriented incidence matrix, viewed as a matrix over Q. The binary cycle space is the kernel of its oriented or unoriented incidence matrix, viewed as a matrix over GF (2). Theorem 1.26. The incidence matrix of an undirected graph G is related to the adja- cency matrix of its line graph L(G) by the following theorem: A(L(G)) = D(G)T D(G) − 2In , where A(L(G)) is the adjacency matrix of the line graph of G. Proof. Let Di denote the ith column of D. Consider the dot product of Di and Dj , i 6= j. The terms contributing to this expression are associated to the vertices which are incident to the ith edge and also to the jth edge. In other words, there is such a vertex (and only one such vertex) if and only if this dot product is equal to 1 if and only if ith edge is incident to the jth edge in G if and only if the vertex in L(G) associated to the ith edge in G is adjacent to the vertex in L(G) associated to the jth edge in G. But this is exactly the condition that the corresponding entry of A(L(G)) is equal to 1. Consider the dot product of Di with itself. The terms contributing to this expression are associated to the vertices which are incident to the ith edge. There are 2 such vertices so this dot product is equal to 2. Subtracting, the 2 in 2In , gives 0, as expect for the diagonal entries of A(L(G)). For a directed graph, the result in the above theorem does not hold in general (except in characteristic 2), as the following example shows. Example 1.27. Consider the graph shown in Figure 1.21, whose line graph is shown in Figure 1.22. sage : G1 = DiGraph ({0:[1 ,2 ,4] , 1:[2 ,3 ,4] , 2:[3 ,4]}) sage : G1 . show () sage : D1 = G1 . incidence_matrix (); D1 [ -1 -1 -1 0 0 0 0 0] [ 0 0 1 -1 -1 -1 0 0] [ 0 1 0 0 0 1 -1 -1] [ 0 0 0 0 1 0 0 1] [ 1 0 0 1 0 0 1 0] sage : A1 = G1 . adjacency_matrix () sage : A1 [0 1 1 0 1] [0 0 1 1 1] [0 0 0 1 1] [0 0 0 0 0] [0 0 0 0 0] sage : G = Graph ({0:[1 ,2 ,4] , 1:[2 ,3 ,4] , 2:[3 ,4]}) sage : D = G . incidence_matrix (); D [ -1 -1 -1 0 0 0 0 0] [ 0 0 1 -1 -1 -1 0 0] [ 0 1 0 0 0 1 -1 -1] [ 0 0 0 0 1 0 0 1] [ 1 0 0 1 0 0 1 0] 26 Chapter 1. Introduction to graph theory sage : D . transpose ()* D [ 2 1 1 1 0 0 1 0] [ 1 2 1 0 0 1 -1 -1] [ 1 1 2 -1 -1 -1 0 0] [ 1 0 -1 2 1 1 1 0] [ 0 0 -1 1 2 1 0 1] [ 0 1 -1 1 1 2 -1 -1] [ 1 -1 0 1 0 -1 2 1] [ 0 -1 0 0 1 -1 1 2] sage : D * D . transpose () [ 3 -1 -1 0 -1] [ -1 4 -1 -1 -1] [ -1 -1 4 -1 -1] [ 0 -1 -1 2 0] [ -1 -1 -1 0 3] sage : ( -1)* G . adjacency_matrix () [ 0 -1 -1 0 -1] [ -1 0 -1 -1 -1] [ -1 -1 0 -1 -1] [ 0 -1 -1 0 0] [ -1 -1 -1 0 0] sage : V = G . vertices () sage : [ G . degree ( v ) for v in V ] [3 , 4 , 4 , 2 , 3] sage : MS8 = MatrixSpace ( QQ , 8 ,8) sage : I8 = MS8 (1) sage : D . transpose ()* D - 2* I8 [ 0 1 1 1 0 0 1 0] [ 1 0 1 0 0 1 -1 -1] [ 1 1 0 -1 -1 -1 0 0] [ 1 0 -1 0 1 1 1 0] [ 0 0 -1 1 0 1 0 1] [ 0 1 -1 1 1 0 -1 -1] [ 1 -1 0 1 0 -1 0 1] [ 0 -1 0 0 1 -1 1 0] sage : LG = G . line_graph () sage : ALG = LG . adjacency_matrix () sage : ALG [0 1 1 1 1 1 0 0] [1 0 1 1 0 0 1 1] [1 1 0 0 0 1 0 1] [1 1 0 0 1 1 1 1] [1 0 0 1 0 1 1 0] [1 0 1 1 1 0 0 1] [0 1 0 1 1 0 0 1] [0 1 1 1 0 1 1 0] sage : G3 = Graph ({0:[1 ,2 ,3 ,6] , 1:[2 ,5 ,6 ,7] , 2:[3 ,4 ,5] ,3:[4] ,4:[5 ,7] ,5:[7]}) # # line graph sage : G3 . adjacency_matrix () [0 1 1 1 0 0 1 0] [1 0 1 0 0 1 1 1] [1 1 0 1 1 1 0 0] [1 0 1 0 1 0 0 0] [0 0 1 1 0 1 0 1] [0 1 1 0 1 0 0 1] [1 1 0 0 0 0 0 0] [0 1 0 0 1 1 0 0] Figure 1.21: A digraph example having 5 vertices and 8 edges. 1.3. Representing graphs in a computer 27 Figure 1.22: The line graph of a digraph example having 5 vertices and 8 edges. Theorem 1.28. The rank (over Q) of the incidence matrix of a directed connected simple graph having n vertices is n − 1. Since G is a simple graph, it has fewer edges than vertices. Proof. Consider the column of D coresponding to e ∈ E. The number of entries equal to +1 is one (corresponding to the vertex at the “head” of e) and the number of entries equal to −1 is also one (corresponding to the vertex at the “tail” of e). All other entries are equal to 0. Therefore, the sum of all the rows in D is the zero vector. This implies rankQ (D) ≤ n − 1. To show that equality is attained, we exhibit n − 1 linearly independent columns of D. Let T be a spanning tree for G. This tree has n − 1 edges and, if you reindex the vertices of G suitably, the columns of D associated to the edges in T are of the form wk = (0, . . . , 0, 1, −1, 0, . . . , 0) ∈ Qn , where the kth entry is a 1 and the (k + 1)st entry is −1 (1 ≤ k ≤ n − 1). These are clearly linearly independent. 1.3.3 Laplacian matrix The degree matrix of a graph G = (V, E) is an n × n diagonal matrix D whose i-th diagonal entry is the degree of the i-th vertex in V . The Laplacian matrix L of G is the difference between the degree matrix and the adjacency matrix: L = D − A. In other words, for an undirected unweighted simple graph, L = [`ij ] is given by −1, if i 6= j and vi vj ∈ E, `ij = di , if i = j, 0, otherwise, 28 Chapter 1. Introduction to graph theory where di = deg(vi ) is the degree of vertex vi . Sage allows you to compute the Laplacian matrix of a graph: sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : G . laplacian_matrix () [ 3 -1 0 -1 0 -1] [ -1 4 -1 -1 -1 0] [ 0 -1 3 -1 0 -1] [ -1 -1 -1 4 -1 0] [ 0 -1 0 -1 2 0] [ -1 0 -1 0 0 2] There are many remarkable properties of the Laplacian matrix. It shall be discussed further in Chapter 5. 1.3.4 Distance matrix Recall that the distance (or geodesic distance) d(v, w) between two vertices v, w ∈ V in a connected graph G = (V, E) is the number of edges in a shortest path connecting them. The n × n matrix [d(vi , vj )] is the distance matrix of G. Sage helps you to compute the distance matrix of a graph: sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : d = [[ G . distance (i , j ) for i in range (1 ,7)] for j in range (1 ,7)] sage : matrix ( d ) [0 1 2 1 2 1] [1 0 1 1 1 2] [2 1 0 1 2 1] [1 1 1 0 1 2] [2 1 2 1 0 3] [1 2 1 2 3 0] The distance matrix is an important quantity which allows one to better understand the “connectivity” of a graph. Distance and connectivity will be discussed in more detail in Chapters 5 and 11. 1.4 Graph transformations 1.5 Isomorphic graphs Determining whether or not two graphs are, in some sense, the “same” is a hard but important problem. Two graphs G and H are isomorphic if there is a bijection f : V (G) → V (H) such that whenever uv ∈ E(G) then f (u)f (v) ∈ E(H). The function f is an isomorphism between G and H. Otherwise, G and H are non-isomorphic. If G and H are isomorphic, we write G ∼ = H. A graph G is isomorphic to a graph H if these two graphs can be labelled in such a way that if u and v are adjacent in G, then their counterparts in V (H) are also adjacent in H. To determine whether or not two graphs are isomorphic is to determine if they are structurally equivalent. Graphs G and H may be drawn differently so that they seem different. However, if G ∼ = H then the isomorphism f : V (G) → V (H) shows that both of these graphs are fundamentally the same. In particular, the order and size of G are equal to those of H, the isomorphism f preserves adjacencies, and deg(v) = deg(f (v)) for all v ∈ G. Since f preserves adjacencies, then adjacencies along a given geodesic path are preserved as well. That is, if v1 , v2 , v3 , . . . , vk is a shortest path between v1 , vk ∈ V (G), then f (v1 ), f (v2 ), f (v3 ), . . . , f (vk ) is a geodesic path between f (v1 ), f (vk ) ∈ V (H). For example, the two graphs in Figure 1.23 are isomorphic to each other. 1.5. Isomorphic graphs 29 (a) (b) Figure 1.23: Two representations of the Franklin graph. e f 1 2 e f c d 3 4 c d a b 5 6 a b (a) C6 (b) G1 (c) G2 Figure 1.24: Isomorphic and nonisomorphic graphs. Example 1.29. Consider the graphs in Figure 1.24. Which pair of graphs are isomor- phic, and which two graphs are non-isomorphic? Solution. If G is a Sage graph, one can use the method G.is_isomorphic() to determine whether or not the graph G is isomorphic to another graph. The following Sage session illustrates how to use G.is_isomorphic(). sage : C6 = Graph ({ " a " :[ " b " ," c " ] , " b " :[ " a " ," d " ] , " c " :[ " a " ," e " ] , \ ... " d " :[ " b " ," f " ] , " e " :[ " c " ," f " ] , " f " :[ " d " ," e " ]}) sage : G1 = Graph ({1:[2 ,4] , 2:[1 ,3] , 3:[2 ,6] , 4:[1 ,5] , \ ... 5:[4 ,6] , 6:[3 ,5]}) sage : G2 = Graph ({ " a " :[ " d " ," e " ] , " b " :[ " c " ," f " ] , " c " :[ " b " ," f " ] , \ ... " d " :[ " a " ," e " ] , " e " :[ " a " ," d " ] , " f " :[ " b " ," c " ]}) sage : C6 . is_isomorphic ( G1 ) True sage : C6 . is_isomorphic ( G2 ) False sage : G1 . is_isomorphic ( G2 ) False Thus, for the graphs C6 , G1 and G2 in Figure 1.24, C6 and G1 are isomorphic, but G1 and G2 are not isomorphic. An important notion in graph theory is the idea of an “invariant”. An invariant is an object f = f (G) associated to a graph G which has the property G∼ = H =⇒ f (G) = f (H). For example, the number of vertices of a graph, f (G) = |V (G)|, is an invariant. 30 Chapter 1. Introduction to graph theory 1.5.1 Adjacency matrices Two n × n matrices A1 and A2 are permutation equivalent if there is a permutation matrix P such that A1 = P A2 P −1 . In other words, A1 is the same as A2 after a suitable re-ordering of the rows and a corresponding re-ordering of the columns. This notion of permutation equivalence is an equivalence relation. To show that two undirected graphs are isomorphic depends on the following result. Theorem 1.30. Consider two directed or undirected graphs G1 and G2 with respective adjacency matrices A1 and A2 . Then G1 and G2 are isomorphic if and only if A1 is permutation equivalent to A2 . This says that the permutation equivalence class of the adjacency matrix is an in- variant. Define an ordering on the set of n×n (0, 1)-matrices as follows: we say A1 < A2 if the list of entries of A1 is less than or equal to the list of entries of A2 in the lexicographical ordering. Here, the list of entries of a (0, 1)-matrix is obtained by concatenating the entries of the matrix, row-by-row. For example, 1 1 1 1 < . 0 1 1 1 Algorithm 1.1 is an immediate consequence of Theorem 1.30. The lexicographically maximal element of the permutation equivalence class of the adjacency matrix of G is called the canonical label of G. Thus, to check if two undirected graphs are isomorphic, we simply check if their canonical labels are equal. This idea for graph isomorphism checking is presented in Algorithm 1.1. Algorithm 1.1 Computing graph isomorphism using canonical labels. Input Two undirected simple graphs G1 and G2 , each having n vertices. Output true if G1 ∼ = G2 ; false otherwise. 1: for i ← 1, 2 do 2: Ai ← adjacency matrix of Gi 3: pi ← permutation equivalence class of Ai 4: A0i ← lexicographically maximal element of pi 5: if A01 = A02 then 6: return true 7: return false 1.5.2 Degree sequence Let G be a graph with n vertices. The degree sequence of G is the ordered n-tuple of the vertex degrees of G arranged in non-increasing order. The degree sequence of G may contain the same degrees, repeated as often as they occur. For example, the degree sequence of C6 is 2, 2, 2, 2, 2, 2 and the degree sequence of the house graph in Figure 1.3 is 3, 3, 2, 2, 2. If n ≥ 3 then the cycle graph Cn has the degree sequence 2, 2, 2, . . . , 2 . | {z } n copies of 2 1.5. Isomorphic graphs 31 The path Pn , for n ≥ 3, has the degree sequence 2, 2, 2, . . . , 2, 1, 1 . | {z } n−2 copies of 2 For positive integer values of n and m, the complete graph Kn has the degree sequence n − 1, n − 1, n − 1, . . . , n − 1 | {z } n copies of n−1 and the complete bipartite graph Km,n has the degree sequence n, n, n, . . . , n, m, m, m, . . . , m . | {z }| {z } m copies of n n copies of m Let S be a non-increasing sequence of non-negative integers. Then S is said to be graphical if it is the degree sequence of some graph. If G is a graph with degree sequence S, we say that G realizes S. Let S = (d1 , d2 , . . . , dn ) be a graphical sequence, P i.e. di ≥ dj for all i ≤ j such that 1 ≤ i, j ≤ n. From Corollary 1.11 we see that di ∈S di = 2k for some integer k ≥ 0. In other words, the sum of a graphical sequence is nonnegative and even. In 1961, Erdős and Gallai [71] used this observation as part of a theorem that provides necessary and sufficient conditions for a sequence to be realized by a simple graph. The result is stated in Theorem 1.31, but the original paper of Erdős and Gallai [71] does not provide an algorithm to construct a simple graph with a given degree sequence. For a simple graph that has a degree sequence with repeated elements, e.g. the degree sequences of Cn , Pn , Kn , and Km,n , it is redundant to verify inequality (1.11) for repeated elements of that sequence. In 2003, Tripathi and Vijay [184] showed that one only needs to verify inequality (1.11) for as many times as there are distinct terms in S. Theorem 1.31. Erdős & Gallai 1961 [71]. Let d = (d1 , d2 , . . . , dn ) be a sequence positive integers such that di ≥ di+1 . Then d is realized by a simple graph if and only of P if i di is even and Xk Xn di ≤ k(k + 1) + min{k, di } (1.11) i=1 j=k+1 for all 1 ≤ k ≤ n − 1. As noted above, Theorem 1.31 is an existence result showing that something ex- ists without providing a construction of the object under consideration. Havel [98] and Hakimi [95,96] independently provided an algorithmic approach that allows for construct- ing a simple graph with a given degree sequence. See Sierksma and Hoogeveen [175] for a coverage of seven criteria for a sequence of integers to be graphic. See Erdős et al. [74] for an extension of the Havel-Hakimi theorem to digraphs. Theorem 1.32. Havel 1955 [98] & Hakimi 1962–3 [95, 96]. Consider the non- increasing sequence S1 = (d1 , d2 , . . . , dn ) of nonnegative integers, where n ≥ 2 and d1 ≥ 1. Then S1 is graphical if and only if the sequence S2 = (d2 − 1, d3 − 1, . . . , dd1 +1 − 1, dd1 +2 , . . . , dn ) is graphical. 32 Chapter 1. Introduction to graph theory Proof. Suppose S2 is graphical. Let G2 = (V2 , E2 ) be a graph of order n − 1 with vertex set V2 = {v2 , v3 , . . . , vn } such that ( di − 1, if 2 ≤ i ≤ d1 + 1, deg(vi ) = di , if d1 + 2 ≤ i ≤ n. Construct a new graph G1 with degree sequence S1 as follows. Add another vertex v1 to V2 and add to E2 the edges v1 vi for 2 ≤ i ≤ d1 + 1. It is clear that deg(v1 ) = d1 and deg(vi ) = di for 2 ≤ i ≤ n. Thus G1 has the degree sequence S1 . On the other hand, suppose S1 is graphical and let G1 be a graph with degree sequence S1 such that (i) The graph G1 has the vertex set V (G1 ) = {v1 , v2 , . . . , vn } and deg(vi ) = di for i = 1, . . . , n. (ii) The degree sum of all vertices adjacent to v1 is a maximum. To obtain a contradiction, suppose v1 is not adjacent to vertices having degrees d2 , d3 , . . . , dd1 +1 . Then there exist vertices vi and vj with dj > di such that v1 vi ∈ E(G1 ) but v1 vj 6∈ E(G1 ). As dj > di , there is a vertex vk such that vj vk ∈ E(G1 ) but vi vk 6∈ E(G1 ). Replacing the edges v1 vi and vj vk with v1 vj and vi vk , respectively, results in a new graph H whose degree sequence is S1 . However, the graph H is such that the degree sum of vertices adjacent to v1 is greater than the corresponding degree sum in G1 , contradicting property (ii) in our choice of G1 . Consequently, v1 is adjacent to d1 other vertices of largest degree. Then S2 is graphical because G1 − v1 has degree sequence S2 . The proof of Theorem 1.32 can be adapted into an algorithm to determine whether or not a sequence of nonnegative integers can be realized by a simple graph. If G is a simple graph, the degree of any vertex in V (G) cannot exceed the order of G. By the handshaking lemma (Theorem 1.9), the sum of all terms in the sequence cannot be odd. Once the sequence passes these two preliminary tests, we then adapt the proof of Theorem 1.32 to successively reduce the original sequence to a smaller sequence. These ideas are summarized in Algorithm 1.2. We now show that Algorithm 1.2 determines whether or not a sequence of integers is realizable by a simple graph. Our input is a sequence S = (d1 , d2 , . . . , dn ) arranged in non-increasing order, where each di ≥ 0. The first test as contained in the if block, otherwise known as a conditional, on line 1 uses the handshaking lemma (Theorem 1.9). During the first run of the while loop, the conditional on line 4 ensures that the sequence S only consists of nonnegative integers. At the conditional on line 6, we know that S is arranged in non-increasing order and has nonnegative integers. If this conditional holds true, then S is a sequence of zeros and it is realizable by a graph with only isolated vertices. Such a graph is simple by definition. The conditional on line 8 uses the following property of simple graphs: If G is a simple graph, then the degree of each vertex of G is less than the order of G. By the time we reach line 10, we know that S has n terms, max(S) > 0, and 0 ≤ di ≤ n − 1 for all i = 1, 2, . . . , n. After applying line 10, S is now a sequence of n − 1 terms with max(S) > 0 and 0 ≤ di ≤ n − 2 for all i = 1, 2, . . . , n − 1. In general, after k rounds of the while loop, S is a sequence of n − k terms with max(S) > 0 1.6. New graphs from old 33 Algorithm 1.2 Havel-Hakimi test for sequences realizable by simple graphs. Input A nonincreasing sequence S = (d1 , d2 , . . . , dn ) of nonnegative integers, where n ≥ 2. Output P true if S is realizable by a simple graph; false otherwise. 1: if i di is odd then 2: return false 3: while true do 4: if min(S) < 0 then 5: return false 6: if max(S) = 0 then 7: return true 8: if max(S) > length(S) − 1 then 9: return false 10: S ← (d2 − 1, d3 − 1, . . . , dd1 +1 − 1, dd1 +2 , . . . , dlength(S) ) 11: sort S in nonincreasing order and 0 ≤ di ≤ n − k − 1 for all i = 1, 2, . . . , n − k. And after n − 1 rounds of the while loop, the resulting sequence has one term whose value is zero. In other words, eventually Algorithm 1.2 produces a sequence with a negative term or a sequence of zeros. 1.5.3 Invariants revisited In some cases, one can distinguish non-isomorphic graphs by considering graph invariants. For instance, the graphs C6 and G1 in Figure 1.24 are isomorphic so they have the same number of vertices and edges. Also, G1 and G2 in Figure 1.24 are non-isomorphic because the former is connected, while the latter is not connected. To prove that two graphs are non-isomorphic, one could show that they have different values for a given graph invariant. The following list contains some items to check off when showing that two graphs are non-isomorphic: 1. the number of vertices, 2. the number of edges, 3. the degree sequence, 4. the length of a geodesic path, 5. the length of the longest path, 6. the number of connected components of a graph. 1.6 New graphs from old This section provides a brief survey of operations on graphs to obtain new graphs from old graphs. Such graph operations include unions, products, edge addition, edge deletion, vertex addition, and vertex deletion. Several of these are briefly described below. 34 Chapter 1. Introduction to graph theory 1.6.1 Union, intersection, and join The disjoint union of graphs is defined as follows. For two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ) with disjoint vertex sets, their disjoint union is the graph G1 ∪ G2 = (V1 ∪ V2 , E1 ∪ E2 ). For example, Figure 1.25 shows the vertex disjoint union of the complete bipartite graph K1,5 with the wheel graph W4 . The adjacency matrix A of the disjoint union of two graphs G1 and G2 is the diagonal block matrix obtained from the adjacency matrices A1 and A2 , respectively. Namely, A1 0 A= . 0 A2 Sage can compute graph unions, as the following example shows. sage : G1 = Graph ({1:[2 ,4] , 2:[1 ,3] , 3:[2 ,6] , 4:[1 ,5] , 5:[4 ,6] , 6:[3 ,5]}) sage : G2 = Graph ({7:[8 ,10] , 8:[7 ,10] , 9:[8 ,12] , 10:[7 ,9] , 11:[10 ,8] , 12:[9 ,7]}) sage : G1u2 = G1 . union ( G2 ) sage : G1u2 . adjacency_matrix () [0 1 0 1 0 0 0 0 0 0 0 0] [1 0 1 0 0 0 0 0 0 0 0 0] [0 1 0 0 0 1 0 0 0 0 0 0] [1 0 0 0 1 0 0 0 0 0 0 0] [0 0 0 1 0 1 0 0 0 0 0 0] [0 0 1 0 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 0 1 0 1] [0 0 0 0 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 0 1 0 1 0 1] [0 0 0 0 0 0 1 1 1 0 1 0] [0 0 0 0 0 0 0 1 0 1 0 0] [0 0 0 0 0 0 1 0 1 0 0 0] In the case where V1 = V2 , then G1 ∪ G2 is simply the graph consisting of all edges in G1 or in G2 . In general, the union of two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ) is defined as G1 ∪ G2 = (V1 ∪ V2 , E1 ∪ E2 ) where V1 ⊆ V2 , V2 ⊆ V1 , V1 = V2 , or V1 ∩ V2 = ∅. Figure 1.26(c) illustrates the graph union where one vertex set is a proper subset of the other. If G1 , G2 , . . . , Gn are the components S of a graph G, then G is obtained by the disjoint union of its components, i.e. G = Gi . Figure 1.25: The vertex disjoint union K1,5 ∪ W4 . The intersection of graphs is defined as follows. For two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ), their intersection is the graph G1 ∩ G2 = (V1 ∩ V2 , E1 ∩ E2 ). Figure 1.26(d) illustrates the intersection of two graphs whose vertex sets overlap. 1.6. New graphs from old 35 3 3 3 3 1 2 1 2 4 4 4 4 1 2 5 6 5 6 1 2 (a) G1 (b) G2 (c) G1 ∪ G2 (d) G1 ∩ G2 Figure 1.26: The union and intersection of graphs with overlapping vertex sets. The symmetric difference of graphs is defined as follows. For two graphs G1 = (V1 , E1 ) and G2 = (V2 , E2 ), their symmetric difference is the graph G1 ∆G2 = (V, E) where V = V1 ∆V2 and the edge set is given by E = (E1 ∆E2 )\{uv | u ∈ V1 ∩ V2 or v ∈ V1 ∩ V2 }. Recall that the symmetric difference of two sets S1 and S2 is defined by S1 ∆S2 = {x ∈ S1 ∪ S2 | x ∈ / S1 ∩ S2 }. In the case where V1 = V2 , then G1 ∆G2 is simply the empty graph. See Figure 1.27 for an illustration of the symmetric difference of two graphs. 5 5 3 9 7 7 9 3 4 1 2 1 2 4 (a) G1 (b) G2 (c) G1 ∆G2 Figure 1.27: The symmetric difference of graphs. The join of two disjoint graphs G1 and G2 , denoted G1 +G2 , is their graph union, with each vertex of one graph connecting to each vertex of the other graph. For example, the join of the cycle graph Cn−1 with a single vertex graph is the wheel graph Wn . Figure 1.28 shows various wheel graphs. 1.6.2 Edge or vertex deletion/insertion Vertex deletion subgraph If G = (V, E) is any graph with at least 2 vertices, then the vertex deletion subgraph is the subgraph obtained from G by deleting a vertex v ∈ V and also all the edges incident to that vertex. The vertex deletion subgraph of G is sometimes denoted G − {v}. Sage can compute vertex deletions, as the following example shows. 36 Chapter 1. Introduction to graph theory (a) W4 (b) W5 (c) W6 (d) W7 (e) W8 (f) W9 Figure 1.28: The wheel graphs Wn for n = 4, . . . , 9. sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : G . vertices () [1 , 2 , 3 , 4 , 5 , 6] sage : E1 = Set ( G . edges ( labels = False )); E1 {(1 , 2) , (4 , 5) , (1 , 4) , (2 , 3) , (3 , 6) , (1 , 6) , (2 , 5) , (3 , 4) , (2 , 4)} sage : E4 = Set ( G . edges_incident ( vertices =[4] , labels = False )); E4 {(4 , 5) , (3 , 4) , (2 , 4) , (1 , 4)} sage : G . delete_vertex (4) sage : G . vertices () [1 , 2 , 3 , 5 , 6] sage : E2 = Set ( G . edges ( labels = False )); E2 {(1 , 2) , (1 , 6) , (2 , 5) , (2 , 3) , (3 , 6)} sage : E1 . difference ( E2 ) == E4 True Figure 1.29 presents a sequence of subgraphs obtained by repeatedly deleting vertices. As the figure shows, when a vertex is deleted from a graph, all edges incident on that vertex are deleted as well. Edge deletion subgraph If G = (V, E) is any graph with at least 1 edge, then the edge deletion subgraph is the subgraph obtained from G by deleting an edge e ∈ E, but not the vertices incident to that edge. The edge deletion subgraph of G is sometimes denoted G − {e}. Sage can compute edge deletions, as the following example shows. sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : E1 = Set ( G . edges ( labels = False )); E1 {(1 , 2) , (4 , 5) , (1 , 4) , (2 , 3) , (3 , 6) , (1 , 6) , (2 , 5) , (3 , 4) , (2 , 4)} sage : V1 = G . vertices (); V1 [1 , 2 , 3 , 4 , 5 , 6] sage : E14 = Set ([(1 ,4)]); E14 {(1 , 4)} sage : G . delete_edge ([1 ,4]) sage : E2 = Set ( G . edges ( labels = False )); E2 1.6. New graphs from old 37 c c b a e a e d d (a) G (b) G − {b} c c e d d (c) G − {a, b} (d) G − {a, b, e} (e) G − {a, b, c, d, e} Figure 1.29: Obtaining subgraphs via repeated vertex deletion. 38 Chapter 1. Introduction to graph theory {(1 , 2) , (4 , 5) , (2 , 3) , (3 , 6) , (1 , 6) , (2 , 5) , (3 , 4) , (2 , 4)} sage : E1 . difference ( E2 ) == E14 True Figure 1.30 shows a sequence of graphs resulting from edge deletion. Unlike vertex deletion, when an edge is deleted the vertices incident on that edge are left intact. b b b c a c a c a (a) G (b) G − {ac} (c) G − {ab, ac, bc} Figure 1.30: Obtaining subgraphs via repeated edge deletion. Vertex cut, cut vertex, or cutpoint A vertex cut (or separating set) of a connected graph G = (V, E) is a subset W ⊆ V such that the vertex deletion subgraph G − W is disconnected. In fact, if v1 , v2 ∈ V are two non-adjacent vertices, then you can ask for a vertex cut W for which v1 , v2 belong to different components of G − W . Sage’s vertex_cut method allows you to compute a minimal cut having this property. For many connected graphs, the removal of a single vertex is sufficient for the graph to be disconnected (see Figure 1.30(c)). Edge cut, cut edge, or bridge If deleting a single, specific edge would disconnect a graph G, that edge is called a bridge. More generally, the edge cut (or disconnecting set or seg) of a connected graph G = (V, E) is a set of edges F ⊆ E whose removal yields an edge deletion subgraph G − F that is disconnected. A minimal edge cut is called a cut set or a bond . In fact, if v1 , v2 ∈ V are two vertices, then you can ask for an edge cut F for which v1 , v2 belong to different components of G − F . Sage’s edge_cut method allows you to compute a minimal cut having this property. For example, any of the three edges in Figure 1.30(c) qualifies as a bridge and those three edges form an edge cut for the graph in question. Theorem 1.33. Let G be a connected graph. An edge e ∈ E(G) is a bridge of G if and only if e does not lie on a cycle of G. Proof. First, assume that e = uv is a bridge of G. Suppose for contradiction that e lies on a cycle C : u, v, w1 , w2 , . . . , wk , u. Then G − e contains a u-v path u, wk , . . . , w2 , w1 , v. Let u1 , v1 be any two vertices in G − e. By hypothesis, G is connected so there is a u1 -v1 path P in G. If e does not lie on P , then P is also a path in G − e so that u1 , v1 are connected, which contradicts our assumption of e being a bridge. On the other hand, if e lies on P , then express P as u1 , . . . , u, v, . . . , v1 or u1 , . . . , v, u, . . . , v1 . 1.6. New graphs from old 39 Now u1 , . . . , u, wk , . . . , w2 , w1 , v, . . . , v1 or u1 , . . . , v, w1 , w2 , . . . , wk , u, . . . , v1 respectively is a u1 -v1 walk in G − e. By Theorem 1.15, G − e contains a u1 -v1 path, which contradicts our assumption about e being a bridge. Conversely, let e = uv be an edge that does not lie on any cycles of G. If G − e has no u-v paths, then we are done. Otherwise, assume for contradiction that G − e has a u-v path P . Then P with uv produces a cycle in G. This cycle contains e, in contradiction of our assumption that e does not lie on any cycles of G. Edge contraction An edge contraction is an operation which, like edge deletion, removes an edge from a graph. However, unlike edge deletion, edge contraction also merges together the two vertices the edge used to connect. For a graph G = (V, E) and an edge uv = e ∈ E, the edge contraction G/e is the graph obtained as follows: 1. Delete the vertices u, v from G. 2. In place of u, v is a new vertex ve . 3. The vertex ve is adjacent to vertices that were adjacent to u, v, or both u and v. The vertex set of G/e = (V 0 , E 0 ) is defined as V 0 = V \{u, v} ∪ {ve } and its edge set is E 0 = wx ∈ E | {w, x} ∩ {u, v} = ∅ ∪ ve w | uw ∈ E\{e} or vw ∈ E\{e} . Make the substitutions E1 = wx ∈ E | {w, x} ∩ {u, v} = ∅ E2 = ve w | uw ∈ E\{e} or vw ∈ E\{e} . Let G be the wheel graph W6 in Figure 1.31(a) and consider the edge contraction G/ab, where ab is the gray colored edge in that figure. Then the edge set E1 denotes all those edges in G each of which is not incident on a, b, or both a and b. These are precisely those edges that are colored red. The edge set E2 means that we consider those edges in G each of which is incident on exactly one of a or b, but not both. The blue colored edges in Figure 1.31(a) are precisely those edges that E2 suggests for consideration. The result of the edge contraction G/ab is the wheel graph W5 in Figure 1.31(b). Figures 1.31(a) to 1.31(f) present a sequence of edge contractions that starts with W6 and repeatedly contracts it to the trivial graph K1 . 1.6.3 Complements The complement of a simple graph has the same vertices, but exactly those edges that are not in the original graph. In other words, if Gc = (V, E c ) is the complement of G = (V, E), then two distinct vertices v, w ∈ V are adjacent in Gc if and only if they are not adjacent in G. We also write the complement of G as G. The sum of the adjacency matrix of G and that of Gc is the matrix with 1’s everywhere, except for 0’s on the main diagonal. A simple graph that is isomorphic to its complement is called a self- complementary graph. Let H be a subgraph of G. The relative complement of G and H is the edge deletion subgraph G − E(H). That is, we delete from G all edges in H. Sage can compute edge complements, as the following example shows. 40 Chapter 1. Introduction to graph theory d d d f e c f e c f a b vab = g e vcg = h (a) G1 (b) G2 = G1 /ab (c) G3 = G2 /cg f e vdh = i e vf i = j vej (d) G4 = G3 /dh (e) G5 = G4 /f i (f) G6 = G5 /ej Figure 1.31: Contracting the wheel graph W6 to the trivial graph K1 . sage : G = Graph ({1:[2 ,4] , 2:[1 ,4] , 3:[2 ,6] , 4:[1 ,3] , 5:[4 ,2] , 6:[3 ,1]}) sage : Gc = G . complement () sage : EG = Set ( G . edges ( labels = False )); EG {(1 , 2) , (4 , 5) , (1 , 4) , (2 , 3) , (3 , 6) , (1 , 6) , (2 , 5) , (3 , 4) , (2 , 4)} sage : EGc = Set ( Gc . edges ( labels = False )); EGc {(1 , 5) , (2 , 6) , (4 , 6) , (1 , 3) , (5 , 6) , (3 , 5)} sage : EG . difference ( EGc ) == EG True sage : EGc . difference ( EG ) == EGc True sage : EG . intersection ( EGc ) {} Theorem 1.34. If G = (V, E) is self-complementary, then the order of G is |V | = 4k or |V | = 4k + 1 for some nonnegative integer k. Furthermore, if n = |V | is the order of G, then the size of G is |E| = n(n − 1)/4. Proof. Let G be a self-complementary graph of order n. Each of G and Gc contains half the number of edges in Kn . From (1.6), we have 1 n(n − 1) n(n − 1) |E(G)| = |E(Gc )| = · = . 2 2 4 Then n | n(n − 1), with one of n and n − 1 being even and the other odd. If n is even, n − 1 is odd so gcd(4, n − 1) = 1, hence by [174, Theorem 1.9] we have 4 | n and so n = 4k for some nonnegative k ∈ Z. If n − 1 is even, use a similar argument to conclude that n = 4k + 1 for some nonnegative k ∈ Z. Theorem 1.35. A graph and its complement cannot be both disconnected. Proof. If G is connected, then we are done. Without loss of generality, assume that G is disconnected and let G be the complement of G. Let u, v be vertices in G. If u, v are in different components of G, then they are adjacent in G. If both u, v belong to 1.6. New graphs from old 41 some component Ci of G, let w be a vertex in a different component Cj of G. Then u, w are adjacent in G, and similarly for v and w. That is, u and v are connected in G and therefore G is connected. 1.6.4 Cartesian product The Cartesian product GH of graphs G and H is a graph such that the vertex set of GH is the Cartesian product V (GH) = V (G) × V (H). Any two vertices (u, u0 ) and (v, v 0 ) are adjacent in GH if and only if either 1. u = v and u0 is adjacent with v 0 in H; or 2. u0 = v 0 and u is adjacent with v in G. The vertex set of GH is V (GH) and the edge set of GH is E(GH) = V (G) × E(H) ∪ E(G) × V (H) . Sage can compute Cartesian products, as the following example shows. sage : Z = graphs . CompleteGraph (2); len ( Z . vertices ()); len ( Z . edges ()) 2 1 sage : C = graphs . CycleGraph (5); len ( C . vertices ()); len ( C . edges ()) 5 5 sage : P = C . cartesian_product ( Z ); len ( P . vertices ()); len ( P . edges ()) 10 15 The path graph Pn is a tree with n vertices V = {v1 , v2 , . . . , vn } and edges E = {(vi , vi+1 ) | 1 ≤ i ≤ n − 1}. In this case, deg(v1 ) = deg(vn ) = 1 and deg(vi ) = 2 for 1 < i < n. The path graph Pn can be obtained from the cycle graph Cn by deleting one edge of Cn . The ladder graph Ln is the Cartesian product of path graphs, i.e. Ln = Pn P1 . The Cartesian product of two graphs G1 and G2 can be visualized as follows. Let V1 = {u1 , u2 , . . . , um } and V2 = {v1 , v2 , . . . , vn } be the vertex sets of G1 and G2 , respectively. Let H1 , H2 , . . . , Hn be n copies of G1 . Place each Hi at the location of vi in G2 . Then ui ∈ V (Hj ) is adjacent to ui ∈ V (Hk ) if and only if vjk ∈ E(G2 ). See Figure 1.32 for an illustration of obtaining the Cartesian product of K3 and P3 . (a) K3 (b) P3 (c) K3 P3 Figure 1.32: The Cartesian product of K3 and P3 . 42 Chapter 1. Introduction to graph theory The hypercube graph Qn is the n-regular graph having vertex set V = (1 , . . . , n ) | i ∈ {0, 1} of cardinality 2n . That is, each vertex of Qn is a bit string of length n. Two vertices v, w ∈ V are connected by an edge if and only if v and w differ in exactly one coordinate.5 The Cartesian product of n edge graphs K2 is a hypercube: (K2 )n = Qn . Figure 1.33 illustrates the hypercube graphs Qn for n = 1, . . . , 4. (a) Q1 (b) Q2 (c) Q3 (d) Q4 Figure 1.33: Hypercube graphs Qn for n = 1, . . . , 4. Example 1.36. The Cartesian product of two hypercube graphs is another hypercube, i.e. Qi Qj = Qi+j . Another family of graphs that can be constructed via Cartesian product is the mesh. Such a graph is also referred to as grid or lattice. The 2-mesh is denoted M (m, n) and is defined as the Cartesian product M (m, n) = Pm Pn . Similarly, the 3-mesh is defined as M (k, m, n) = Pk Pm Pn . In general, for a sequence a1 , a2 , . . . , an of n > 0 positive integers, the n-mesh is given by M (a1 , a2 , . . . , an ) = Pa1 Pa2 · · · Pan where the 1-mesh is simply the path graph M (k) = Pk for some positive integer k. Figure 1.34(a) illustrates the 2-mesh M (3, 4) = P3 P4 , while the 3-mesh M (3, 2, 3) = P3 P2 P3 is presented in Figure 1.34(b). 5 In other words, the Hamming distance between v and w is equal to 1. 1.7. Problems 43 (a) M (3, 4) (b) M (3, 2, 3) Figure 1.34: The 2-mesh M (3, 4) and the 3-mesh M (3, 2, 3). 1.6.5 Graph minors A graph H is called a minor of a graph G if H is isomorphic to a graph obtained by a sequence of edge contractions on a subgraph of G. The order in which a sequence of such contractions is performed on G does not affect the resulting graph H. A graph minor is not in general a subgraph. However, if G1 is a minor of G2 and G2 is a minor of G3 , then G1 is a minor of G3 . Therefore, the relation “being a minor of” is a partial ordering on the set of graphs. For example, the graph in Figure 1.31(c) is a minor of the graph in Figure 1.31(a). The following non-intuitive fact about graph minors was proven by Neil Robertson and Paul Seymour in a series of 20 papers spanning 1983 to 2004. This result is known by various names including the Robertson-Seymour theorem, the graph minor theorem, or Wagner’s conjecture (named after Klaus Wagner). Theorem 1.37. Robertson & Seymour 1983–2004. If an infinite list G1 , G2 , . . . of finite graphs is given, then there always exist two indices i < j such that Gi is a minor of Gj . Many classes of graphs can be characterized by forbidden minors: a graph belongs to the class if and only if it does not have a minor from a certain specified list. We shall see examples of this in Chapter ??. 1.7 Problems A problem left to itself dries up or goes rotten. But fertilize a problem with a solution— you’ll hatch out dozens. — N. F. Simpson, A Resounding Tinkle, 1958 1.1. For each graph in Figure 1.7, do the following: (a) Construct the graph using Sage. (b) Find its adjacency matrix. (c) Find its node and edge sets. (d) How many nodes and edges are in the graph? 44 Chapter 1. Introduction to graph theory Alice Bob Carol Figure 1.35: Graph representation of a social network. (e) If applicable, find all of each node’s in-coming and out-going edges. Hence find the node’s indegree and outdegree. 1.2. In the friendship network of Figure 1.35, Carol is a mutual friend of Alice and Bob. How many possible ways are there to remove exactly one edge such that, in the resulting network, Carol is no longer a mutual friend of Alice and Bob? Karlsruhe 197 54 Augsburg Mannheim 57 72 Munich 383 Kassel 145 Frankfurt 149 97 Nuremberg 90 Würzburg 157 154 Stuttgart Erfurt Figure 1.36: Graph representation of a routing network. 1.3. The routing network of German cities in Figure 1.36 shows that each pair of distinct 1.7. Problems 45 cities are connected by a flight path. The weight of each edge is the flight distance in kilometers between the two corresponding cities. In particular, there is a flight path connecting Karlsruhe and Stuttgart. What is the shortest route between Karlsruhe and Stuttgart? Suppose we can remove at least one edge from this network. How many possible ways are there to remove edges such that, in the resulting network, Karlsruhe is no longer connected to Stuttgart via a flight path? 1.4. Let D = (V, E) be a digraph of size q. Show that X X id(v) = od(v) = q. v∈V v∈V 1.5. If G is a simple graph of order n > 0, show that deg(v) < n for all v ∈ V (G). 1.6. Let G be a graph of order n and size m. Then G is called an overfull graph if m > ∆(G) · bn/2c. If m = ∆(G) · bn/2c + 1, then G is said to be just overfull. It can be shown that overfull graphs have odd order. Equivalently, let G be of odd order. We can define G to be overfull if m > ∆(G) · (n − 1)/2, and G is just overfull if m = ∆(G) · (n − 1)/2 + 1. Find an overfull graph and a graph that is just overfull. Some basic results on overfull graphs are presented in [52]. 1.7. Fix a positive integer n and denote by Γ(n) the number of simple graphs on n vertices. Show that n Γ(n) = 2( 2 ) = 2n(n−1)/2 . 1.8. Let G be an undirected graph whose unoriented incidence matrix is Mu and whose oriented incidence matrix is Mo . (a) Show that the sum of the entries in any row of Mu is the degree of the corresponding vertex. (b) Show that the sum of the entries in any column of Mu is equal to 2. (c) If G has no self-loops, show that each column of Mo sums to zero. 1.9. Let G be a loopless digraph and let M be its incidence matrix. (a) If r is a row of M , show that the number of occurrences of −1 in r counts the outdegree of the vertex corresponding to r. Show that the number of occurrences of 1 in r counts the indegree of the vertex corresponding to r. (b) Show that each column of M sums to 0. 1.10. Let G be a digraph and let M be its incidence matrix. For any row r of M , let m be the frequency of −1 in r, let p be the frequency of 1 in r, and let t be twice the frequency of 2 in r. If v is the vertex corresponding to r, show that the degree of v is deg(v) = m + p + t. 1.11. Let G be an undirected graph without self-loops and let M and its oriented in- cidence matrix. Show that the Laplacian matrix L of G satisfies L = M × M T , where M T is the transpose of M . 46 Chapter 1. Introduction to graph theory 1.12. Let J1 denote the incidence matrix of G1 and let J2 denote the incidence matrix of G2 . Find matrix theoretic criteria on J1 and J2 which hold if and only if G1 ∼ = G2 . In other words, find the analog of Theorem 1.30 for incidence matrices. 1.13. Show that the complement of an edgeless graph is a complete graph. 1.14. Let GH be the Cartesian product of two graphs G and H. Show that |E(GH)| = |V (G)| · |E(H)| + |E(G)| · |V (H)|. Figure 1.37: Euler’s polygon division problem for the hexagon. 1.15. In 1751, Leonhard Euler posed a problem to Christian Goldbach, a problem that now bears the name “Euler’s polygon division problem”. Given a plane convex polygon having n sides, how many ways are there to divide the polygon into tri- angles using only diagonals? For our purposes, we consider only regular polygons having n sides for n ≥ 3 and any two diagonals must not cross each other. For example, the triangle is a regular 3-gon, the square a regular 4-gon, the pentagon a regular 5-gon, etc. In the case of the hexagon considered as the cycle graph C6 , there are 14 ways to divide it into triangles, as shown in Figure 1.37, resulting in 14 graphs. However, of those 14 graphs only 3 are nonisomorphic to each other. (a) What is the number of ways to divide a pentagon into triangles using only diagonals? List all such divisions. If each of the resulting so divided pentagons is considered a graph, how many of those graphs are nonisomorphic to each other? (b) Repeat the above exercise for the heptagon. (c) Let En be the number of ways to divide an n-gon into triangles using only diagonals. For n ≥ 1, the Catalan numbers Cn are defined as 1 2n Cn = . n+1 n 1.7. Problems 47 Dörrie [64, pp.21–27] showed that En is related to the Catalan numbers via the equation En = Cn−1 . Show that 1 2n + 2 Cn = . 4n + 2 n + 1 For k ≥ 2, show that the Catalan numbers satisfy the recurrence relation 4k − 2 Ck = Ck−1 . k+1 1.16. A graph is said to be planar if it can be drawn on the plane in such a way that no two edges cross each other. For example, the complete graph Kn is planar for n = 1, 2, 3, 4, but K5 is not planar (see Figure 1.13). Draw a planar version of K4 as presented in Figure 1.13(b). Is the graph in Figure 1.9 planar? For n = 1, 2, . . . , 5, enumerate all simple nonisomorphic graphs on n vertices that are planar; only work with undirected graphs. 1.17. If n ≥ 3, show that the join of Cn and K1 is the wheel graph Wn+1 . In other words, show that Cn + K1 = Wn+1 . 1.18. A common technique for generating “random” numbers is the linear congruential method, a generalization of the Lehmer generator [132] introduced in 1949. First, we choose four integers: m, modulus, 0<m a, multiplier, 0≤a<m c, increment, 0≤c<m X0 , seed, 0 ≤ X0 < m where the value X0 is also referred to as the starting value. Then iterate the relation Xn+1 = (aXn + c) mod m, n≥0 and halt when the relation produces the seed X0 or when it produces an integer Xk such that Xk = Xi for some 0 ≤ i < k. The resulting sequence S = (X0 , X1 , . . . , Xn ) is called a linear congruential sequence. Define a graph theoretic representation of S as follows: let the vertex set be V = {X0 , X1 , . . . , Xn } and let the edge set be E = {Xi Xi+1 | 0 ≤ i < n}. The resulting graph G = (V, E) is called the linear congruential graph of the linear congruential sequence S. See chapter 3 of [122] for other techniques for generating “random” numbers. (a) Compute the linear congruential sequences Si with the following parameters: (i) S1 : m = 10, a = c = X0 = 7 (ii) S2 : m = 10, a = 5, c = 7, X0 = 0 (iii) S3 : m = 10, a = 3, c = 7, X0 = 2 (iv) S4 : m = 10, a = 2, c = 5, X0 = 3 48 Chapter 1. Introduction to graph theory (b) Let Gi be the linear congruential graph of Si . Draw each of the graphs Gi . Draw the graph resulting from the union [ Gi . i (c) Let m, a, c, and X0 be the parameters of a linear congruential sequence where (i) c is relatively prime to m; (ii) b = a − 1 is a multiple of p for each prime p that divides m; and (iii) 4 divides b if 4 divides m. Show that the corresponding linear congruential graph is the wheel graph Wm on m vertices. 1.19. We want to generate a random bipartite graph whose first and second partitions have n1 and n2 vertices, respectively. Describe and present pseudocode to generate the required random bipartite graph. What is the worst-case runtime of your algorithm? Modify your algorithm to account for a third parameter m that specifies the number of edges in the resulting bipartite graph. 1.20. Describe and present pseudocode to generate a random regular graph. What is the worst-case runtime of your algorithm? 1.21. The Cantor-Schröder-Bernstein theorem states that if A, B are sets and we have an injection f : A → B and an injection g : B → A, then there is a bijection between A and B, thus proving that A and B have the same cardinality. Here we use bipartite graphs and other graph theoretic concepts to prove the Cantor-Schröder-Bernstein theorem. The full proof can be found in [201]. (a) Is it possible for A and B to be bipartitions of V and yet satisfy A ∩ B 6= ∅? (b) Now assume that A ∩ B = ∅ and define a bipartite graph G = (V, E) with A and B being the two partitions of V , where for any x ∈ A and y ∈ B we have xy ∈ E if and only if either f (x) = y or g(y) = x. Show that deg(v) = 1 or deg(v) = 2 for each v ∈ V . (c) Let C be a component of G and let A0 ⊆ A and B 0 ⊆ B contain all vertices in the component C. Show that |A0 | = |B 0 |. 1.22. Fermat’s little theorem states that if p is prime and a is an integer not divisible by p, then p divides ap − a. Here we cast the problem within the context of graph theory and prove it using graph theoretic concepts. The full proof can be found in [99, 201]. (a) Let G = (V, E) be a graph with V being the set of all sequences (a1 , a2 , . . . , ap ) of integers 1 ≤ ai ≤ a and aj 6= ak for some j 6= k. Show that G has ap − a vertices. (b) Define the edge set of G as follows. If u, v ∈ V such that u = (u1 , u2 , . . . , up ) and v = (up , u1 , . . . , up−1 ), then uv ∈ E. Show that each component of G is a cycle of length p. (c) Show that G has (ap − a)/p components. 1.7. Problems 49 1.23. For the finite automaton in Figure ??, identify the following: (a) The states set Q. (b) The alphabet set Σ. (c) The transition function δ : Q × Σ → Q. (d) The initial state q0 ∈ Q. (e) The set of final states F ⊆ Q. 3 2 3 2 3 2 4 1 4 1 4 1 5 0 5 0 5 0 6 9 6 9 6 9 7 8 7 8 7 8 (a) (b) (c) Figure 1.38: Various 3-regular graphs on 10 vertices. (a) k = 4 (b) k = 6 (c) k = 8 Figure 1.39: Various k-circulant graphs for k = 4, 6, 8. 1.24. The cycle graph Cn is a 2-regular graph. If 2 < r < n/2, unlike the cycle graph there are various realizations of an r-regular graph; see Figure 1.38 for the case of r = 3 and n = 10. The k-circulant graph on n vertices can be considered as an intermediate graph between Cn and a k-regular graph. Let k and n be positive integers satisfying k < n/2 with k being even. Suppose G = (V, E) is a simple undirected graph with vertex set V = {0, 1, . . . , n − 1}. Define the edge set of G as follows. Each i ∈ V is incident with each of i + j mod n and i − j mod n for j ∈ {1, 2, . . . , k/2}. With the latter edge set, G is said to be a k-circulant graph, a type of graphs used in constructing small-world networks (see section 11.4). Refer to Figure 1.39 for examples of k-circulant graphs. (a) Describe and provide pseudocode of an algorithm to construct a k-circulant graph on n vertices. (b) Show that the cycle graph Cn is 2-circulant. 50 Chapter 1. Introduction to graph theory (c) Show that the sum of all degrees of a k-circulant graph on n vertices is nk. (d) Show that a k-circulant graph is k-regular. (e) Let C be the collection of all k-regular graphs on n vertices. If each k-regular graph from C is equally likely to be chosen, what is the probability that a k-circulant graph be chosen from C? Chapter 2 Trees and forests — Randall Munroe, xkcd, http://xkcd.com/71/ In section 1.2.1, we briefly touched upon trees and provided examples of how trees could be used to model hierarchical structures. This chapter provides an in-depth study of trees, their properties, and various applications. After defining trees and related con- cepts in section 2.1, we then present various basic properties of trees in section 2.2. Each connected graph G has an underlying subgraph called a spanning tree that con- tains all the vertices of G. Spanning trees are discussed in section 2.3 together with various common algorithms for finding spanning trees. We then discuss binary trees in section 2.4, followed by an application of binary trees to coding theory in section 2.5. Whereas breadth- and depth-first searches are general methods for traversing a graph, trees require specialized techniques in order to visit their vertices, a topic that is taken up in section 2.6. 2.1 Definitions and examples I think that I shall never see A poem lovely as a tree. — Joyce Kilmer, Trees and Other Poems, 1914, “Trees” Recall that a path in a graph G = (V, E) whose start and end vertices are the same is called a cycle. We say G is acyclic, or a forest, if it has no cycles. In a forest, a vertex of degree one is called an endpoint or a leaf . Any vertex that is not a leaf is called an 51 52 Chapter 2. Trees and forests internal vertex. A connected forest is a tree. In other words, a tree is a graph without cycles and each edge is a bridge. A forest can also be considered as a collection of trees. A rooted tree T is a tree with a specified root vertex v0 , i.e. exactly one vertex has been specially designated as the root of T . However, if G is a rooted tree with root vertex v0 having degree one, then by convention we do not call v0 an endpoint or a leaf. The depth depth(v) of a vertex v in T is its distance from the root. The height height(T ) of T is the length of a longest path starting from the root vertex, i.e. the height is the maximum depth among all vertices of T . It follows by definition that depth(v) = 0 if and only if v is the root of T , height(T ) = 0 if and only if T is the trivial graph, depth(v) ≥ 0 for all v ∈ V (T ), and height(T ) ≤ diam(T ). The Unix, in particular Linux, filesystem hierarchy can be viewed as a tree (see Figure 2.1). As shown in Figure 2.1, the root vertex is designated with the forward slash, which is also referred to as the root directory. Other examples of trees include the organism classification tree in Figure 2.2, the family tree in Figure 2.3, and the expression tree in Figure 2.4. A directed tree is a digraph which would be a tree if the directions on the edges were ignored. A rooted tree can be regarded as a directed tree since we can imagine an edge uv for u, v ∈ V being directed from u to v if and only if v is further away from v0 than u is. If uv is an edge in a rooted tree, then we call v a child vertex with parent u. Directed trees are pervasive in theoretical computer science, as they are useful structures for describing algorithms and relationships between objects in certain datasets. / bin etc home lib opt proc tmp usr ... anne sam ... bin include local share src ... acyclic diff dot gc neato ... Figure 2.1: The Linux filesystem hierarchy. An ordered tree is a rooted tree for which an ordering is specified for the children of each vertex. An n-ary tree is a rooted tree for which each vertex that is not a leaf has at most n children. The case n = 2 are called binary trees. An n-ary tree is said to be complete if each of its internal vertices has exactly n children and all leaves have the same depth. A spanning tree of a connected, undirected graph G is a subgraph that is a tree and containing all vertices of G. Example 2.1. Consider the 4 × 4 grid graph with 16 vertices and 24 edges. Two examples of a spanning tree are given in Figure 2.5 by using a darker line shading for its edges. Example 2.2. For n = 1, . . . , 6, how many distinct (nonisomorphic) trees are there of order n? Construct all such trees for each n. 2.1. Definitions and examples 53 organism plant animal tree flower invertebrate vetebrate deciduous evergreen bird mammal finch rosella sparrow dolphin human whale Figure 2.2: Classification tree of organisms. Nikolaus senior Jacob Nicolaus Johann Nicolaus I Nicolaus II Daniel Johann II Johann III Daniel II Jakob II Figure 2.3: Bernoulli family tree of mathematicians. + × × × a a 2 a b b b Figure 2.4: Expression tree for the perfect square a2 + 2ab + b2 . 54 Chapter 2. Trees and forests (a) (b) Figure 2.5: Two spanning trees for the 4 × 4 grid graph. Solution. For n = 1, there is only one tree of order 1, i.e. K1 . The same is true for n = 2 and n = 3, where the required trees are P2 and P3 , respectively (see Figure 2.6). We have two trees of order n = 4 (see Figure 2.7), three of order n = 5 (see Figure 2.8), and six of order n = 6 (see Figure 2.9). (a) n = 1 (b) n = 2 (c) n = 3 Figure 2.6: All distinct trees of order n = 1, 2, 3. (a) (b) Figure 2.7: All distinct trees of order n = 4. Example 2.3. Let T = (V, E) be a tree with vertex set V = {a, b, c, d, e, f, v, w, x, y, z} edge set E = {va, vw, wx, wy, xb, xc, yd, yz, ze, zf } and root vertex v. Verify that T is a binary tree. Suppose that x is the root of the branch we want to remove from T . Find all children of x and cut off the branch rooted at x from T . Is the resulting graph also a binary tree? 2.1. Definitions and examples 55 (a) (b) (c) Figure 2.8: All distinct trees of order n = 5. (a) (b) (c) (d) (e) (f) Figure 2.9: All distinct trees of order n = 6. 56 Chapter 2. Trees and forests Solution. We construct the tree T in Sage as follows: sage : T = DiGraph ({ ... " v " :[ " a " ," w " ] , " w " :[ " x " ," y " ] , ... " x " :[ " c " ," b " ] , " y " :[ " z " ," d " ] , ... " z " :[ " f " ," e " ]}) sage : for v in T . vertex_iterator (): ... print ( v ) , a c b e d f w v y x z sage : for e in T . edge_iterator (): ... print ( " % s % s " % ( e [0] , e [1])) , wy wx va vw yd yz xc xb ze zf Each vertex in a binary tree has at most 2 children. Use this definition to test whether or not a graph is a binary tree. sage : T . is_tree () True sage : def is_bintree1 ( G ): ... for v in G . vertex_iterator (): ... if len ( G . neighbors_out ( v )) > 2: ... return False ... return True sage : is_bintree1 ( T ) True Here’s another way to test for binary trees. Let T be an undirected rooted tree. Each vertex in a binary tree has a maximum degree of 3. If the root vertex is the only vertex with degree 2, then T is a binary tree. (Problem 2.5 asks you to prove this result.) We can use this test because the root vertex v of T is the only vertex with two children. sage : def is_bintree2 ( G ): ... if G . is_tree () and max ( G . degree ()) == 3 and G . degree (). count (2) == 1: ... return True ... return False sage : is_bintree2 ( T . to_undirected ()) True As x is the root vertex of the branch we want to cut off from T , we could use breadth- or depth-first search to determine all the children of x. We then delete x and its children from T . sage : T2 = copy ( T ) sage : # using breadth - first search sage : V = list ( T . breadth_first_search ( " x " )); V [ ’x ’ , ’c ’ , ’b ’] sage : T . delete_vertices ( V ) sage : for v in T . vertex_iterator (): ... print ( v ) , a e d f w v y z sage : for e in T . edge_iterator (): ... print ( " % s % s " % ( e [0] , e [1])) , wy va vw yd yz ze zf sage : # using depth - first search sage : V = list ( T2 . depth_first_search ( " x " )); V [ ’x ’ , ’b ’ , ’c ’] sage : T2 . delete_vertices ( V ) sage : for v in T2 . vertex_iterator (): ... print ( v ) , a e d f w v y z sage : for e in T2 . edge_iterator (): ... print ( " % s % s " % ( e [0] , e [1])) , wy va vw yd yz ze zf The resulting graph T is a binary tree because each vertex has at most two children. sage : T Digraph on 8 vertices sage : is_bintree1 ( T ) True Notice that the test defined in the function is_bintree2 can no longer be used to test whether or not T is a binary tree, because T now has two vertices, i.e. v and w, each of 2.1. Definitions and examples 57 which has degree 2. Consider again the organism classification tree in Figure 2.2. We can view the vertex “organism” as the root of the tree and having two children. The first branch of “or- ganism” is the subtree rooted at “plant” and its second branch is the subtree rooted at “animal”. We form the complete tree by joining an edge between “organism” and “plant”, and an edge between “organism” and “animal”. The subtree rooted at “plant” can be constructed in the same manner. The first branch of this subtree is the subtree rooted at “tree” and the second branch is the subtree rooted at “flower”. To construct the subtree rooted at “plant”, we join an edge between “plant” and “tree”, and an edge between “plant” and “flower”. The other subtrees of the tree in Figure 2.2 can be constructed using the above recursive procedure. In general, the recursive construction in Theorem 2.4 provides an alternative way to define trees. We say construction because it provides an algorithm to construct a tree, as opposed to the nonconstructive definition presented earlier in this section, where we defined the conditions under which a graph qualifies as a tree without presenting a procedure to construct a tree. Furthermore, we say recursive since a larger tree can be viewed as being constructed from smaller trees, i.e. join up existing trees to obtain a new tree. The recursive construction of trees as presented in Theorem 2.4 is illustrated in Figure 2.10. Theorem 2.4. Recursive construction of trees. An isolated vertex is a tree. That single vertex is the root of the tree. Given a collection T1 , T2 , . . . , Tn of n > 0 trees, construct a new tree as follows: 1. Let T be a tree having exactly the one vertex v, which is the root of T . 2. Let vi be the root of the tree Ti . 3. For i = 1, 2, . . . , n, add the edge vvi to T and add Ti to T . That is, each vi is now a child of v. The result is the tree T rooted at v with vertex set ! [ V (T ) = {v} ∪ V (Ti ) i and edge set [ E(T ) = {vvi } ∪ E(Ti ) . i The following game is a variant of the Shannon switching game, due to Edmonds and Lehman. We follow the description in Oxley’s survey [158]. Recall that a minimal edge cut of a graph is also called a bond of the graph. The following two-person game is played on a connected graph G = (V, E). Two players Alice and Bob alternately tag elements of E. Alice’s goal is to tag the edges of a spanning tree, while Bob’s goal is to tag the edges of a bond. If we think of this game in terms of a communication network, then Bob’s goal is to separate the network into pieces that are no longer connected to each other, while Alice is aiming to reinforce edges of the network to prevent their destruction. Each move for Bob consists of destroying one edge, while each move for Alice involves securing an edge against destruction. The next result characterizes winning strategies on G. The full proof can be found in Oxley [158]. See Rasmussen [164] for optimization algorithms for solving similar games. 58 Chapter 2. Trees and forests v ... T1 T2 Tn Figure 2.10: Recursive construction of a tree. Theorem 2.5. The following statements are equivalent for a connected graph G = (V, E). 1. Bob plays first and Alice can win against all possible strategies of Bob. 2. The graph G has 2 edge-disjoint spanning trees. 3. For all partitions P of the vertex set V , the number of edges of G that join vertices in different classes of the partition is at least 2(|P | − 1). 2.2 Properties of trees All theory, dear friend, is grey, but the golden tree of actual life springs ever green. — Johann Wolfgang von Goethe, Faust, part 1, 1808 By Theorem 1.33, each edge of a tree is a bridge. Removing any edge of a tree partitions the tree into two components, each of which is a subtree of the original tree. The following results provide further basic characterizations of trees. Theorem 2.6. Any tree T = (V, E) has size |E| = |V | − 1. Proof. This follows by induction on the number of vertices. By definition, a tree has no cycles. We need to show that any tree T = (V, E) has size |E| = |V | − 1. For the base case |V | = 1, there are no edges. Assume for induction that the result holds for all integers less than or equal to k ≥ 2. Let T = (V, E) be a tree having k + 1 vertices. Remove an edge from T , but not the vertices it is incident to. This disconnects T into two components T1 = (V1 , E1 ) and T2 = (V2 , E2 ), where |E| = |E1 | + |E2 | + 1 and |V | = |V1 | + |V2 | (and possibly one of the Ei is empty). Each Ti is a tree satisfying the conditions of the induction hypothesis. Therefore, |E| = |E1 | + |E2 | + 1 = |V1 | − 1 + |V2 | − 1 + 1 = |V | − 1. as required. 2.2. Properties of trees 59 Corollary 2.7. If T = (V, E) is a graph of order |V | = n, then the following are equivalent: 1. T is a tree. 2. T contains no cycles and has n − 1 edges. 3. T is connected and has n − 1 edges. 4. Every edge of T is a cut set. Proof. (1) =⇒ (2): This holds by definition of trees and Theorem 2.6. (2) =⇒ (3): If T = (V, E) has k connected components then it is a disjoint union of trees Ti = (Vi , Ei ), i = 1, 2, . . . , k, for some k. By part (2), each of these satisfy |Ei | = |Vi | − 1 so k X |E| = |Ei | i=1 k X = |Vi | − k i=1 = |V | − k. This contradicts part (2) unless k = 1. Therefore, T is connected. (3) =⇒ (4): If removing an edge e ∈ E leaves T = (V, E) connected then T 0 = (V, E 0 ) is a tree, where E 0 = E−e. However, this means that |E 0 | = |E|−1 = |V |−1−1 = |V | − 2, which contradicts part (3). Therefore e is a cut set. (4) =⇒ (1): From part (2) we know that T has no cycles and from part (3) we know that T is connected. Conclude by the definition of trees that T is a tree. Theorem 2.8. Let T = (V, E) be a tree and let u, v ∈ V be distinct vertices. Then T has exactly one u-v path. Proof. Suppose for contradiction that P : v0 = u, v1 , v2 , . . . , vk = v and Q : w0 = u, w1 , w2 , . . . , w` = v are two distinct u-v paths. Then P and Q has a common vertex x, which is possibly x = u. For some i ≥ 0 and some j ≥ 0 we have vi = x = wj , but vi+1 6= wj+1 . Let y be the first vertex after x such that y belongs to both P and Q. (It is possible that y = v.) We now have two distinct x-y paths that have only x and y in common. Taken together, these two x-y paths result in a cycle, contradicting our hypothesis that T is a tree. Therefore T has only one u-v path. Theorem 2.9. If T = (V, E) is a graph then the following are equivalent: 60 Chapter 2. Trees and forests 1. T is a tree. 2. For any new edge e, the join T + e has exactly one cycle. Proof. (1) =⇒ (2): Let e = uv be a new edge connecting u, v ∈ V . Suppose that P : v0 = w, v1 , v2 , . . . , vk = w and P 0 : v00 = w, v10 , v20 , . . . , v`0 = w are two cycles in T + e. If either P or P 0 does not contain e, say P does not contain e, then P is a cycle in T . Let u = v0 and let v = v1 . The edge (v0 = w, v1 ) is a u-v path and the sequence v = v1 , v2 , . . . , vk = w = u taken in reverse order is another u-v path. This contradicts Theorem 2.8. We may now suppose that P and P 0 both contain e. Then P contains a subpath P0 = P − e (which is not closed) that is the same as P except it lacks the edge from u to v. Likewise, P 0 contains a subpath P00 = P 0 − e (which is not closed) that is the same as P 0 except it lacks the edge from u to v. By Theorem 2.8, these u-v paths P0 and P00 must be the same. This forces P and P 0 to be the same, which proves part (2). (2) =⇒ (1): Part (2) implies that T is acyclic. (Otherwise, it is trivial to make two cycles by adding an extra edge.) We must show T is connected. Suppose T is disconnected. Let u be a vertex in one component, T1 say, of T and v a vertex in another component, T2 say, of T . Adding the edge e = uv does not create a cycle (if it did then T1 and T2 would not be disjoint), which contradicts part (2). Taking together the results in this section, we have the following characterizations of trees. Theorem 2.10. Basic characterizations of trees. If T = (V, E) is a graph with n vertices, then the following statements are equivalent: 1. T is a tree. 2. T contains no cycles and has n − 1 edges. 3. T is connected and has n − 1 edges. 4. Every edge of T is a cut set. 5. For any pair of distinct vertices u, v ∈ V , there is exactly one u-v path. 6. For any new edge e, the join T + e has exactly one cycle. Let G = (V1 , E1 ) be a graph and T = (V2 , E2 ) a subgraph of G that is a tree. As in part (6) of Theorem 2.10, we see that adding just one edge in E1 − E2 to T will create a unique cycle in G. Such a cycle is called a fundamental cycle of G. The set of such fundamental cycles of G depends on T . The following result essentially says that if a tree has at least one edge, then the tree has at least two vertices each of which has degree one. In other words, each tree of order ≥ 2 has at least two pendants. Theorem 2.11. Every nontrivial tree has at least two leaves. 2.2. Properties of trees 61 Proof. Let T be a nontrivial tree of order m and size n. Consider the degree sequence d1 , d2 , . . . , dm of T where d1 ≤ d2 ≤ · · · ≤ dm . As T is nontrivial and connected, then m ≥ 2 and di ≥ 1 for i = 1, 2, . . . , m. If T has less than two leaves, then d1 ≥ 1 and di ≥ 2 for 2 ≤ i ≤ m, hence m X di ≥ 1 + 2(m − 1) = 2m − 1. (2.1) i=1 But by Theorems 1.9 and 2.6, we have m X di = 2n = 2(m − 1) = 2m − 2 i=1 which contradicts inequality (2.1). Conclude that T has at least two leaves. Theorem 2.12. If T is a tree of order m and G is a graph with minimum degree δ(G) ≥ m − 1, then T is isomorphic to a subgraph of G. Proof. Use an inductive argument on the number of vertices. The result holds for m = 1 because K1 is a subgraph of every nontrivial graph. The result also holds for m = 2 since K2 is a subgraph of any graph with at least one edge. Let m ≥ 3, let T1 be a tree of order m − 1, and let H be a graph with δ(H) ≥ m − 2. Assume for induction that T1 is isomorphic to a subgraph of H. We need to show that if T is a tree of order m and G is a graph with δ(G) ≥ m − 1, then T is isomorphic to a subgraph of G. Towards that end, consider a leaf v of T and let u be a vertex of T such that u is adjacent to v. Then T − v is a tree of order m − 1 and δ(G) ≥ m − 1 > m − 2. Apply the inductive hypothesis to see that T − v is isomorphic to a subgraph T 0 of G. Let u0 be the vertex of T 0 that corresponds to the vertex u of T under an isomorphism. Since deg(u0 ) ≥ m − 1 and T 0 has m − 2 vertices distinct from u0 , it follows that u0 is adjacent to some w ∈ V (G) such that w ∈ / V (T 0 ). Therefore T is isomorphic to the graph obtained by adding the edge u0 w to T 0 . Example 2.13. Consider a positive integer n. The Euler phi function ϕ(n) counts the number of integers a, with 1 ≤ a ≤ n, such that gcd(a, n) = 1. The Euler phi sequence of n is obtained by repeatedly iterating ϕ(n) with initial iteration value n. Continue on iterating and stop when the output of ϕ(αk ) is 1, for some positive integer αk . The number of terms generated by the iteration, including the initial iteration value n and the final value of 1, is the length of ϕ(n). (a) Let s0 = n, s1 , s2 , . . . , sk = 1 be the Euler phi sequence of n and produce a digraph G of this sequence as follows. The vertex set of G is V = {s0 = n, s1 , s2 , . . . , sk = 1} and the edge set of G is E = {si si+1 | 0 ≤ i < k}. Produce the digraphs of the Euler phi sequences of 15, 22, 33, 35, 69, and 72. Construct the union of all such digraphs and describe the resulting graph structure. (b) For each n = 1, 2, . . . , 1000, compute the length of ϕ(n) and plot the pairs (n, ϕ(n)) on one set of axes. Solution. The Euler phi sequence of 15 is 15, ϕ(15) = 8, ϕ(8) = 4, ϕ(4) = 2, ϕ(2) = 1. 62 Chapter 2. Trees and forests The Euler phi sequences of 22, 33, 35, 69, and 72 can be similarly computed to obtain their respective digraph representations. The union of all such digraphs is a directed tree rooted at 1, as shown in Figure 2.11(a). Figure 2.11(b) shows a scatterplot of n versus the length of ϕ(n). 1 2 12 4 10 8 length of ϕ(n) 8 10 6 15 20 24 22 4 33 44 35 72 2 0 0 200 400 600 800 1,000 69 n (a) (b) Figure 2.11: Union of digraphs of Euler phi sequences and scatterplot. 2.3 Minimum spanning trees Suppose we want to design an electronic circuit connecting several components. If these components represent the vertices of a graph and a wire connecting two components represents an edge of the graph, then for economical reasons we will want to connect the components together using the least amount of wire. The problem essentially amounts to finding a minimum spanning tree in the graph containing these vertices. But what is a spanning tree? We can characterize a spanning tree in several ways, each leading to an algorithm for constructing a spanning tree. Let G be a connected graph and let T be a subgraph of G. If T is a tree that contains all the vertices of G, then T is called a spanning tree of G. We can think of T as a tree that is also an edge-deletion subgraph of G. That is, we start with a connected graph G and delete an edge from G such that the resulting edge-deletion subgraph T1 is still connected. If T1 is a tree, then we have obtained a spanning tree of G. Otherwise, we delete an edge from T1 to obtain an edge-deletion subgraph T2 that is still connected. If T2 is a tree, then we are done. Otherwise, we repeat the above procedure until we obtain an edge-deletion subgraph Tk of G such that Tk is connected, Tk is a tree, and it contains all vertices of G. Each edge removal does not decrease the number of vertices and must also leave 2.3. Minimum spanning trees 63 the resulting edge-deletion subgraph connected. Thus eventually the above procedure results in a spanning tree of G. Our discussion is summarized in Algorithm 2.1. Algorithm 2.1 Randomized spanning tree construction. Input A connected graph G. Output A spanning tree of G. 1: T ← G 2: while T is not a tree do 3: e ← random edge of T 4: if T − e is connected then 5: T ←T −e 6: return T Another characterization of a spanning tree T of a connected graph G is that T is a maximal set of edges of G that contains no cycle. Kruskal’s algorithm (see section 2.3.1) exploits this condition to construct a minimum spanning tree (MST). A minimum span- ning tree is a spanning tree of a weighted graph having lowest total weight among all possible spanning trees of the graph. A third characterization of a spanning tree is that it is a minimal set of edges that connect all vertices, a characterization that results in yet another algorithm called Prim’s algorithm (see section 2.3.2) for constructing mini- mum spanning trees. The task of determining a minimum spanning tree in a connected weighted graph is called the minimum spanning tree problem. As early as 1926, Otakar Borůvka stated [34, 35] this problem and offered a solution now known as Borůvka’s algorithm (see section 2.3.3). See [88, 139] for a history of the minimum spanning tree problem. 2.3.1 Kruskal’s algorithm In 1956, Joseph B. Kruskal published [125] a procedure for constructing a minimum span- ning tree of a connected weighted graph G = (V, E). Now known as Kruskal’s algorithm, with a suitable implementation the procedure runs in O |E| · log |E| time. Variants of Kruskal’s algorithm include the algorithm by Prim [163] and that by Loberman and Weinberger [137]. Kruskal’s algorithm belongs to the class of greedy algorithms. As will be explained below, when constructing a minimum spanning tree Kruskal’s algorithm considers only the edge having minimum weight among all available edges. Given a weighted nontrivial graph G = (V, E) that is connected, let w : E → R be the weight function of G. The first stage is creating a “skeleton” of the tree T that is initially set to be a graph without edges, i.e. T = (V, ∅). The next stage involves sorting the edges of G by weights in nondecreasing order. In other words, we label the edges of G as follows: E = {e1 , e2 , . . . , en } where n = |E| and w(e1 ) ≤ w(e2 ) ≤ · · · ≤ w(en ). Now consider each edge ei for i = 1, 2, . . . , n. We add ei to the edge set of T provided that ei does not result in T having a cycle. The only way adding ei = ui vi to T would create a cycle is if both ui and vi were endpoints of edges (not necessarily distinct) in the same connected component of T . As long as the acyclic condition holds with the addition of a new edge to T , we add that new edge. Following the acyclic test, we also test that the (updated) graph 64 Chapter 2. Trees and forests T is a tree of G. As G is a graph of order |V |, apply Theorem 2.10 to see that if T has size |V | − 1, then it is a spanning tree of G. Algorithm 2.2 provides pseudocode of our discussion of Kruskal’s algorithm. When the algorithm halts, it returns a minimum spanning tree of G. The correctness of Algorithm 2.2 is proven in Theorem 2.14. Algorithm 2.2 Kruskal’s algorithm. Input A connected weighted graph G = (V, E) with weight function w. Output A minimum spanning tree of G. 1: m ← |V | 2: T ← ∅ 3: sort E = {e1 , e2 , . . . , en } by weights so that w(e1 ) ≤ w(w2 ) ≤ · · · ≤ w(en ) 4: for i ← 1, 2, . . . , n do 5: if ei ∈/ E(T ) and T ∪ {ei } is acyclic then 6: T ← T ∪ {ei } 7: if |T | = m − 1 then 8: return T Theorem 2.14. Correctness of Algorithm 2.2. If G is a nontrivial connected weighted graph, then Algorithm 2.2 outputs a minimum spanning tree of G. Proof. Let G be a nontrivial connected graph of order m and having weight function w. Let T be a subgraph of G produced by Kruskal’s algorithm 2.2. By construction, T is a spanning tree of G with E(T ) = {e1 , e2 , . . . , em−1 } where w(e1 ) ≤ w(e2 ) ≤ · · · ≤ w(em−1 ) so that the total weight of T is m−1 X w(T ) = w(ei ). i=1 Suppose for contradiction that T is not a minimum spanning tree of G. Among all the minimum spanning trees of G, let H be a minimum spanning tree of G such that H has the most number of edges in common with T . As T and H are distinct subgraphs of G, then T has at least an edge not belonging to H. Let ei ∈ E(T ) be the first edge not in H. Construct the graph G0 = H + ei obtained by adding the edge ei to H. Note that G0 has exactly one cycle C. Since T is acyclic, there exists an edge e0 ∈ E(C) such that e0 is not in T . Construct the graph T0 = G0 − e0 obtained by deleting the edge e0 from G0 . Then T0 is a spanning tree of G with w(T0 ) = w(H) + w(ei ) − w(e0 ) and w(H) ≤ w(T0 ) and hence w(e0 ) ≤ w(ei ). By Kruskal’s algorithm 2.2, ei is an edge of minimum weight such that {e1 , e2 , . . . , ei−1 } ∪ {ei } is acyclic. Furthermore, the subgraph {e1 , e2 , . . . , ei−1 , e0 } of H is acyclic. Thus we have w(ei ) = w(e0 ) and w(T0 ) = w(H) and so T is a minimum spanning tree of G. By construction, T0 has more edges in common with T than H has with T , in contradiction of our hypothesis. 2.3. Minimum spanning trees 65 def kruskal ( G ): """ Implements Kruskal ’s algorithm to compute a MST of a graph . INPUT : G - a connected edge - weighted graph or digraph whose vertices are assumed to be 0 , 1 , .... , n -1. OUTPUT : T - a minimum weight spanning tree . If G is not explicitly edge - weighted then the algorithm assumes all edge weights are 1. The tree T returned is a weighted graph , even if G is not . EXAMPLES : sage : A = matrix ([[0 ,1 ,2 ,3] ,[0 ,0 ,2 ,1] ,[0 ,0 ,0 ,3] ,[0 ,0 ,0 ,0]]) sage : G = DiGraph (A , format = " adjacency_matrix " , weighted = True ) sage : TE = kruskal ( G ); TE . edges () [(0 , 1 , 1) , (0 , 2 , 2) , (1 , 3 , 1)] sage : G . edges () [(0 , 1 , 1) , (0 , 2 , 2) , (0 , 3 , 3) , (1 , 2 , 2) , (1 , 3 , 1) , (2 , 3 , 3)] sage : G = graphs . PetersenGraph () sage : TE = kruskal ( G ); TE . edges () [(0 , 1 , 1) , (0 , 4 , 1) , (0 , 5 , 1) , (1 , 2 , 1) , (1 , 6 , 1) , (2 , 3 , 1) , (2 , 7 , 1) , (3 , 8 , 1) , (4 , 9 , 1)] TODO : Add ’’ verbose ’’ option to make steps more transparent . ( Useful for teachers and students .) """ T_vertices = G . vertices () # a list of the form range ( n ) T_edges = [] E = G . edges () # a list of triples # start ugly hack Er = [ list ( x ) for x in E ] E0 = [] for x in Er : x . reverse () E0 . append ( x ) E0 . sort () E = [] for x in E0 : x . reverse () E . append ( tuple ( x )) # end ugly hack to get E is sorted by weight for x in E : # find edges of T TV = flatten ( T_edges ) u = x [0] v = x [1] if not ( u in TV and v in TV ): T_edges . append ([ u , v ]) # find adj mat of T if G . weighted (): AG = G . w e i g h t e d _ a dj a c e n c y _ m a t r i x () else : AG = G . adjacency_matrix () GV = G . vertices () n = len ( GV ) AT = [] for i in GV : rw = [0]* n for j in GV : if [i , j ] in T_edges : rw [ j ] = AG [ i ][ j ] AT . append ( rw ) AT = matrix ( AT ) return Graph ( AT , format = " adjacency_matrix " , weighted = True ) Here is an example. We start with the grid graph. This is implemented in Sage such that the vertices are given by the coordinates of the grid the graph lies on, as opposed to 0, 1, . . . , n − 1. Since the above implementation of Kruskal’s algorithm assumes that the vertices are V = {0, 1, . . . , n − 1}, we first redefine the graph suitable for running Kruskal’s algorithm on it. sage : G = graphs . GridGraph ([4 ,4]) 66 Chapter 2. Trees and forests sage : A = G . adjacency_matrix () sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) sage : T = kruskal ( G ); T . edges () [(0 , 1 , 1) , (0 , 4 , 1) , (1 , 2 , 1) , (1 , 5 , 1) , (2 , 3 , 1) , (2 , 6 , 1) , (3 ,7 , 1) , (4 , 8 , 1) , (5 , 9 , 1) , (6 , 10 , 1) , (7 , 11 , 1) , (8 , 12 , 1) , (9 , 13 , 1) , (10 , 14 , 1) , (11 , 15 , 1)] An illustration of this graph is given in Figure 2.12. Figure 2.12: Kruskal’s algorithm for the 4 × 4 grid graph. 2.3.2 Prim’s algorithm Like Kruskal’s algorithm, Prim’s algorithm uses a greedy approach to computing a min- imum spanning tree of a connected weighted graph G = (V, E), where n = |V | and m = |E|. The algorithm was developed in 1930 by Czech mathematician V. Jarnı́k [107] and later independently by R. C. Prim [163] and E. W. Dijkstra [61]. However, Prim was the first to present an implementation that runs in time O(n2 ). Using 2-heaps, the runtime can be reduced [118] to O(m log n). With a Fibonacci heap implementation [82], the runtime can be reduced even further to O(m + n log n). Pseudocode of Prim’s algorithm is given in Algorithm 2.3. For each v ∈ V , cost[v] denotes the minimum weight among all edges connecting v to a vertex in the tree T , and parent[v] denotes the parent of v in T . During the algorithm’s execution, vertices v that are not in T are organized in the minimum-priority queue Q, prioritized according to cost[v]. Lines 1 to 3 set each cost[v] to a number that is larger than any weight in the graph G, usually written ∞. The parent of each vertex is set to NULL because we have not yet started constructing the MST T . In lines 4 to 6, we choose an arbitrary vertex r from V and mark that vertex as the root of T . The minimum-priority queue is set to be all vertices from V . We set cost[r] to zero, making r the only vertex so far with a cost that is < ∞. During the first execution of the while loop from lines 7 to 12, r is the first vertex to be extracted from Q and processed. Line 8 extracts a vertex u from Q based on the key cost, thus moving u to the vertex set of T . Line 9 considers all vertices adjacent to u. In an undirected graph, these are simply the neighbors of u. (In a digraph, one could try to replace adj(u) with the out-neighbors oadj(u). Unfortunately, in the digraph case the Prim algorithm in general fails to find a minimum spanning tree with the same orientation as the original digraph.) The while loop updates the cost and parent fields of each vertex v adjacent to u that is not in T . If parent[v] 6= NULL, then cost[v] < ∞ and cost[v] is the weight of an edge connecting v to some vertex already in T . Lines 13 to 14 construct the edge set of the minimum spanning tree and return this edge set. The proof of correctness of Algorithm 2.3 is similar to the proof of Theorem 2.14. 2.3. Minimum spanning trees 67 Figure 2.13 shows the minimum spanning tree rooted at vertex 1 as a result of running Prim’s algorithm over a digraph; Figure 2.14 shows the corresponding tree rooted at vertex 5 of an undirected graph. Algorithm 2.3 Prim’s algorithm. Input A weighted connected graph G = (V, E) with weight function w. Output A minimum spanning tree T of G. 1: for each v ∈ V do 2: cost[v] ← ∞ 3: parent[v] ← Null 4: r ← arbitrary vertex of V 5: cost[r] ← 0 6: Q ← V 7: while Q 6= ∅ do 8: u ← extractMin(Q) 9: for each v ∈ adj(u) do 10: if v ∈ Q and w(u, v) < cost[v] then 11: parent[v] ← u 12: cost[v] ← w(u, v) 13: T ← (v, parent[v]) | v ∈ V − {r} 14: return T def prim ( G ): """ Implements Prim ’s algorithm to compute a MST of a graph . INPUT : G - a connected graph . OUTPUT : T - a minimum weight spanning tree . REFERENCES : http :// en . wikipedia . org / wiki / Prim ’ s_algorithm """ T_vertices = [0] # assumes G . vertices = range ( n ) T_edges = [] E = G . edges () # a list of triples V = G . vertices () # start ugly hack to sort E Er = [ list ( x ) for x in E ] E0 = [] for x in Er : x . reverse () E0 . append ( x ) E0 . sort () E = [] for x in E0 : x . reverse () E . append ( tuple ( x )) # end ugly hack to get E is sorted by weight for x in E : u = x [0] v = x [1] if u in T_vertices and not ( v in T_vertices ): T_edges . append ([ u , v ]) T_vertices . append ( v ) # found T_vertices , T_edges # find adj mat of T if G . weighted (): AG = G . w e i g h t e d _ a dj a c e n c y _ m a t r i x () else : AG = G . adjacency_matrix () GV = G . vertices () 68 Chapter 2. Trees and forests n = len ( GV ) AT = [] for i in GV : rw = [0]* n for j in GV : if [i , j ] in T_edges : rw [ j ] = AG [ i ][ j ] AT . append ( rw ) AT = matrix ( AT ) return Graph ( AT , format = " adjacency_matrix " , weighted = True ) sage : A = matrix ([[0 ,1 ,2 ,3] , [3 ,0 ,2 ,1] , [2 ,1 ,0 ,3] , [1 ,1 ,1 ,0]]) sage : G = DiGraph (A , format = " adjacency_matrix " , weighted = True ) sage : E = G . edges (); E [(0 , 1 , 1) , (0 , 2 , 2) , (0 , 3 , 3) , (1 , 0 , 3) , (1 , 2 , 2) , (1 , 3 , 1) , (2 , 0 , 2) , (2 , 1 , 1) , (2 , 3 , 3) , (3 , 0 , 1) , (3 , 1 , 1) , (3 , 2 , 1)] sage : prim ( G ) Multi - graph on 4 vertices sage : prim ( G ). edges () [(0 , 1 , 1) , (0 , 2 , 2) , (1 , 3 , 1)] 1 1 1 2 1 2 3 1 3 1 2 2 0 1 1 2 0 1 1 2 2 2 3 1 3 1 1 3 1 3 3 3 (a) Original digraph. (b) 1st iteration of while loop. 1 1 1 2 3 1 2 0 1 1 2 0 1 2 2 3 1 1 1 3 1 3 3 (c) 2nd iteration of while loop. (d) Final MST. Figure 2.13: Running Prim’s algorithm over a digraph. sage : A = matrix ([[0 ,7 ,0 ,5 ,0 ,0 ,0] , [0 ,0 ,8 ,9 ,7 ,0 ,0] , [0 ,0 ,0 ,0 ,5 ,0 ,0] , \ ... [0 ,0 ,0 ,0 ,15 ,6 ,0] , [0 ,0 ,0 ,0 ,0 ,8 ,9] , [0 ,0 ,0 ,0 ,0 ,0 ,11] , [0 ,0 ,0 ,0 ,0 ,0 ,0]]) 2.3. Minimum spanning trees 69 sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) sage : E = G . edges (); E [(0 , 1 , 7) , (0 , 3 , 5) , (1 , 2 , 8) , (1 , 3 , 9) , (1 , 4 , 7) , (2 , 4 , 5) , (3 , 4 , 15) , (3 , 5 , 6) , (4 , 5 , 8) , (4 , 6 , 9) , (5 , 6 , 11)] sage : prim ( G ). edges () [(0 , 1 , 7) , (0 , 3 , 5) , (1 , 2 , 8) , (1 , 4 , 7) , (3 , 5 , 6) , (4 , 6 , 9)] 2.3.3 Borůvka’s algorithm Borůvka’s algorithm [34, 35] is a procedure for finding a minimum spanning tree in a weighted connected graph G = (V, E) for which all edge weights are distinct. It was first published in 1926 by Otakar Borůvka but subsequently rediscovered by many others, including Choquet [53] and Florek et al. [77]. If G has order n = |V | and size m = |E|, it can be shown that Borůvka’s algorithm runs in time O(m log n). Algorithm 2.4 Borůvka’s algorithm. Input A weighted connected graph G = (V, E) with weight function w. All the edge weights of G are distinct. Output A minimum spanning tree T of G. 1: n ← |V | 2: T ← K n 3: while |E(T )| < n − 1 do 4: for each component T 0 of T do 5: e0 ← edge of minimum weight that leaves T 0 6: E(T ) ← E(T ) ∪ e0 7: return T Algorithm 2.4 provides pseudocode of Borůvka’s algorithm. Given a weighted con- nected graph G = (V, E) all of whose edge weights are distinct, the initialization steps in lines 1 and 2 construct a spanning forest T of G, i.e. the subgraph of G containing all of the latter’s vertices and no edges. The initial forest has n components, each being the trivial graph K1 . The while loop from lines 3 to 6 constructs a spanning tree of G via a recursive procedure similar to Theorem 2.4. For each component T 0 of T , we consider all the out-going edges of T 0 and choose an edge e0 that has minimum weight among all such edges. This edge is then added to the edge set of T . In this way, two distinct components, each of which is a tree, are joined together by a bridge. At the end of the while loop, our final graph is a minimum spanning tree of G. Note that the forest-merging steps in the for loop from lines 4 to 6 are amenable to parallelization, hence the alternative name to Borůvka’s algorithm: the parallel forest-merging method. Example 2.15. Figure 2.15 illustrates the gradual construction of a minimum spanning tree for the undirected graph given in Figure 2.15(a). In this case, we require two iterations of the while loop in Borůvka’s algorithm in order to obtain the final minimum spanning tree in Figure 2.15(d). def which_index (x , L ): """ L is a list of sublists ( or tuple of sets or list of tuples , etc ). Returns the index of the first sublist which x belongs to , or None if x is not in flatten ( L ). 70 Chapter 2. Trees and forests 6 6 11 11 5 9 5 9 8 8 6 4 6 4 15 5 15 5 3 7 2 3 7 2 9 8 9 8 5 1 5 1 7 7 0 0 (a) Original undirected graph. (b) 1st iteration of while loop. 6 6 11 11 5 9 5 9 8 8 6 4 6 4 15 5 15 5 3 7 2 3 7 2 9 8 9 8 5 1 5 1 7 7 0 0 (c) 2nd iteration of while loop. (d) 3rd iteration of while loop. 6 6 11 5 9 5 9 8 6 4 6 4 15 5 5 3 7 2 3 7 2 9 8 5 1 5 1 7 7 0 0 (e) 4th iteration of while loop. (f) Final MST. Figure 2.14: Running Prim’s algorithm over an undirected graph. 2.3. Minimum spanning trees 71 6 6 8 7 10 3 4 2 3 2 9.5 2 4 5 3 5 4 5 6 9 11 0 1 1 0 1 (a) Original undirected graph. (b) 0th iteration of while loop. 6 6 8 7 10 7 10 3 4 2 3 4 2 2 2 4 5 4 5 0 1 1 0 1 1 (c) 1st iteration of while loop. (d) 2nd iteration of while loop. Figure 2.15: Recursive construction of MST via Borůvka’s algorithm. 72 Chapter 2. Trees and forests The 0 - th element in Lx = [ L . index ( S ) for S in L if x in S ] almost works , but if the list is empty then Lx [0] throws an exception . EXAMPLES : sage : L = [[1 ,2 ,3] ,[4 ,5] ,[6 ,7 ,8]] sage : which_index (3 , L ) 0 sage : which_index (4 , L ) 1 sage : which_index (7 , L ) 2 sage : which_index (9 , L ) sage : which_index (9 , L ) == None True """ for S in L : if x in S : return L . index ( S ) return None def boruvka ( G ): """ Implements Boruvka ’s algorithm to compute a MST of a graph . INPUT : G - a connected edge - weighted graph with distinct weights . OUTPUT : T - a minimum weight spanning tree . REFERENCES : http :// en . wikipedia . org / wiki / Boruvka ’ s_algorithm """ T_vertices = [] # assumes G . vertices = range ( n ) T_edges = [] T = Graph () E = G . edges () # a list of triples V = G . vertices () # start ugly hack to sort E Er = [ list ( x ) for x in E ] E0 = [] for x in Er : x . reverse () E0 . append ( x ) E0 . sort () E = [] for x in E0 : x . reverse () E . append ( tuple ( x )) # end ugly hack to get E is sorted by weight for e in E : # create about | V |/2 edges of T " cheaply " TV = T . vertices () if not ( e [0] in TV ) or not ( e [1] in TV ): T . add_edge ( e ) for e in E : # connect the " cheapest " components to get T C = T . c o n n e c t e d _ c o m p o n e n t s _ s u b g r a p h s () VC = [ S . vertices () for S in C ] if not ( e in T . edges ()) and ( which_index ( e [0] , VC ) != which_index ( e [1] , VC )): if T . is_connected (): break T . add_edge ( e ) return T Some examples using Sage: sage : A = matrix ([[0 ,1 ,2 ,3] , [4 ,0 ,5 ,6] , [7 ,8 ,0 ,9] , [10 ,11 ,12 ,0]]) sage : G = DiGraph (A , format = " adjacency_matrix " , weighted = True ) sage : boruvka ( G ) Multi - graph on 4 vertices sage : boruvka ( G ). edges () [(0 , 1 , 1) , (0 , 2 , 2) , (0 , 3 , 3)] sage : A = matrix ([[0 ,2 ,0 ,5 ,0 ,0 ,0] , [0 ,0 ,8 ,9 ,7 ,0 ,0] , [0 ,0 ,0 ,0 ,1 ,0 ,0] ,\ ... [0 ,0 ,0 ,0 ,15 ,6 ,0] , [0 ,0 ,0 ,0 ,0 ,3 ,4] , [0 ,0 ,0 ,0 ,0 ,0 ,11] , [0 ,0 ,0 ,0 ,0 ,0 ,0]]) sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) 2.3. Minimum spanning trees 73 sage : E = G . edges (); E [(0 , 1 , 2) , (0 , 3 , 5) , (1 , 2 , 8) , (1 , 3 , 9) , (1 , 4 , 7) , (2 , 4 , 1) , (3 , 4 , 15) , (3 , 5 , 6) , (4 , 5 , 3) , (4 ,6 , 4) , (5 , 6 , 11)] sage : boruvka ( G ) Multi - graph on 7 vertices sage : boruvka ( G ). edges () [(0 , 1 , 2) , (0 , 3 , 5) , (2 , 4 , 1) , (3 , 5 , 6) , (4 , 5 , 3) , (4 , 6 , 4)] sage : A = matrix ([[0 ,1 ,2 ,5] , [0 ,0 ,3 ,6] , [0 ,0 ,0 ,4] , [0 ,0 ,0 ,0]]) sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) sage : boruvka ( G ). edges () [(0 , 1 , 1) , (0 , 2 , 2) , (2 , 3 , 4)] sage : A = matrix ([[0 ,1 ,5 ,0 ,4] , [0 ,0 ,0 ,0 ,3] , [0 ,0 ,0 ,2 ,0] , [0 ,0 ,0 ,0 ,0] , [0 ,0 ,0 ,0 ,0]]) sage : G = Graph (A , format = " adjacency_matrix " , weighted = True ) sage : boruvka ( G ). edges () [(0 , 1 , 1) , (0 , 2 , 5) , (1 , 4 , 3) , (2 , 3 , 2)] 2.3.4 Circuit matrix Recall, the term cycle refers to a closed path. If G is a digraph then a cycle refers to a sequence of edges which form a path in the associated undirected graph. If repeated vertices are allowed, it is more often called a closed walk. If the path is a simple path, with no repeated vertices or edges other than the starting and ending vertices, it may also be called a simple cycle or circuit. A cycle in a directed graph is called a directed cycle. Let Z1 , . . . , ZM denote an enumeration of the cycles (simple closed paths) of G. The cycle matrix or circuit matrix is a M × m matrix C = (cij ) whose rows are parameterized by the cycles and whose columns are parameterized by the edges E = {e1 , . . . , em }, where 1, ei ∈ Zi , cij = 0, otherwise. Recall the incidence matrix was defined in §1.3.2. Theorem 2.16. If G is a directed graph then the rows of the incidence matrix D(G) are orthogonal to the rows of C(G). Proof. We first show that C · Dt = 0. Consider the ith row of C and jth column of Dt . There are non-zero entries in the corresponding entries of these vectors if and only if the jth vertex of G is incident to an edge which occurs in the ith cycle of G. Assume this entry is non-zero. Since G is a directed graph, there is another vertex (the other vertex defining this edge) for which the associated entry is the same but with opposite sign. Therefore the dot product of the ith row of Q and jth column of Dt is zero. The theorem above implies that the column space of the matrix C t (namely, the image of the associated linear transformation) is contained in the kernel of the incidence matrix D. Let F be a field. There is a general result called Sylvester’s Law of Nullity which says that if K is an r × s matrix over F and L is an s × t matrix over F (so KL is defined), and if KL = 0, then rankF (K) + rankF (L) ≤ s. 74 Chapter 2. Trees and forests (This is an immediate corollary of the rank plus nullity theorem from linear algebra.) It follows from this fact that rankQ (D) + rankQ (C) ≤ m. As a corollary to Theorem 1.28, we see that rankQ (D) ≤ m − n + 1. Theorem 2.17. If G is a directed graph then rankQ (C) = m − n + 1. Proof. To show that equality is attained, we exhibit m − n + 1 linearly independent rows of C. Let T = (V (T ), E(T )) be a spanning tree for G. This tree has n − 1 edges and so G has m − n + 1 edges not in T . Let e ∈ / E(T ) be such an edge. The graph T + e is a cycle. Recall such a cycle is called a fundamental cycle (associated to T ). There are exactly m − n + 1 such cycles. Note that no other fundamental cycle “supports” edge e. Suppose that e is the ith edge in E(G). The row of C associated to the cycle T + e has a 1 in the ith coordinate. Every other row of C associated to a fundamental cycle has a 0 in the ith coordinate. Therefore, the rows of C associated to these m − n + 1 fundamental cycles are linearly independent. 2.3.5 Cutset matrix The term cutset (or edge cutset) refers to a sequence of edges in a connected graph G = (V, E) which, when removed from G, creates a disconnected graph and, furthermore, that no proper subset of these edges has this property. When a graph can be disconnected by removing only one edge, that edge is called a bridge. A cutset can be associated to each vertex v ∈ V as follows. Lemma 2.18. The subset Sv ⊂ E of all the edges incident to v forms forms a cutset. This uses only the definitions and is left to the interested reader as an exercise. Let S1 , . . . , SN denote the cutsets of G. The cutset matrix N × m matrix Q = (qij ) whose rows are parameterized by the cutsets and whose columns are parameterized by the edges E = {e1 , . . . , em }, where 1, ei ∈ Si , qij = 0, otherwise. Theorem 2.19. If G is a connected digraph then rankQ (Q) = n − 1. The analog of this theorem for undirected graphs is false (as the examples below show). However, the undirected analog does work if you use the rank over GF (2). Proof. Let G be any connected graph (directed or not). By Lemma 2.18 above, for each v ∈ V there is a cut-set Sv consisting of all edges incident to v. The characteristic vector qv = (q1 , . . . , qm ) of this set (qi = 1 if ei ∈ Sv and qi = 0 otherwise) gives us a row 2.3. Minimum spanning trees 75 vector in the incidence matrix and in the cut-set matrix. Moreover, all such rows in the incidence matrix are of this form, so rankQ (Q) ≥ rankQ (C) = n − 1. Suppose now G is a directed graph. We show equality must hold. Let S be any edge cutset in G and, for each v ∈ V , let Sv denote the associated cutset as above. If S defined a partitioning V = V1 ∪ V2 then X X S= Sv = Sv , v∈V1 v∈V2 where the sum of two cutsets is simply the formal sum as oriented edges in the free abelian group Z[E]. From this it is clear that the dimension of the row-span of Q is equal to the dimension of the row-span of the submatrix of Q given by the Sv ’s. Example 2.20. To construct the example, we will make use of the following Python function. def ed ge _cutset_matrix ( G ): """ Returns the edge cutset matrix of the connected graph $ G $ . """ V = G . vertices () E = G . edges () rows = [] for v1 in V : for v2 in V : if v1 != v2 : S = G . edge_cut ( v1 , v2 , value_only = False )[1] char_S = lambda e : int ( bool ( e in S )) rows . append ([ char_S ( e ) for e in E ]) Q = matrix ( rows ) return Q We use the function above in the examples below. For the cube graph in three dimensions (see Figure 2.16) and for the Desargues graph (see Figure 2.17), the undirected analog of Theorem 2.19 does hold. sage : G = graphs . CubeGraph (3) sage : G 3 - Cube : Graph on 8 vertices sage : Q = edge_cutset_matrix ( G ) sage : rank ( Q ) 7 sage : G = graphs . DesarguesGraph () sage : G Desargues Graph : Graph on 20 vertices sage : e dg e_cutset_matrix ( G ). rank () 19 On the other hand, for the Frucht graph (see Figure 2.18): sage : G = graphs . FruchtGraph () sage : G Frucht graph : Graph on 12 vertices sage : e dg e_cutset_matrix ( G ). rank () 12 These shall be discussed further in §9.1. 76 Chapter 2. Trees and forests Figure 2.16: The 3-dimensional cube graph. Figure 2.17: The Desargues graph. 2.4. Binary trees 77 Figure 2.18: The Frucht graph. 2.4 Binary trees A binary tree is a rooted tree with at most two children per parent. Each child is designated as either a left-child or a right-child . Thus binary trees are also 2-ary trees. Some examples of binary trees are illustrated in Figure 2.19. Given a vertex v in a binary tree T of height h, the left subtree of v is comprised of the subtree that spans the left-child of v and all of this child’s descendants. The notion of a right-subtree of a binary tree is similarly defined. Each of the left and right subtrees of v is itself a binary tree with height ≤ h − 1. If v is the root vertex, then each of its left and right subtrees has height ≤ h − 1, and at least one of these subtrees has height equal to h − 1. (a) (b) (c) (d) Figure 2.19: Examples of binary trees. Theorem 2.21. If T is a complete binary tree of height h, then T has 2h+1 − 1 vertices. 78 Chapter 2. Trees and forests Proof. Argue by induction on h. The assertion of the theorem is trivially true in the base case h = 0. Let k ≥ 0 and assume for induction that any complete binary tree of height k has order 2k+1 − 1. Suppose T is a complete binary tree of height k + 1 and denote the left and right subtrees of T by T1 and T2 , respectively. Each Ti (for i = 1, 2) is a complete binary tree of height k and by our induction hypothesis Ti has 2k+1 − 1 vertices. Thus T has order 1 + (2k+1 − 1) + (2k+1 − 1) = 2k+2 − 1 as required. Theorem 2.21 provides a useful upper bound on the order of a binary tree of a given height. This upper bound is stated in the following corollary. Corollary 2.22. A binary tree of height h has at most 2h+1 − 1 vertices. We now count the number of possible binary trees on n vertices. Let bn be the number of binary trees of order n. For n = 0, we set b0 = 1. The trivial graph is the only binary tree with one vertex, hence b1 = 1. Suppose n > 1 and let T be a binary tree on n vertices. Then the left subtree of T has order 0 ≤ i ≤ n − 1 and the right subtree has n − 1 − i vertices. As there are bi possible left subtrees and bn−1−i possible right subtrees, T has a total of bi bn−1−i different combinations of left and right subtrees. Summing from i = 0 to i = n − 1 and we have n−1 X bn = bi bn−1−i . (2.2) i=0 Expression (2.2) is known as the Catalan recursion and the number bn is the n-th Catalan number, which we know from problem 1.15 can be expressed in the closed form 1 2n bn = . (2.3) n+1 n Figures 2.20 to 2.22 enumerate all the different binary trees on 2, 3, and 4 vertices, respectively. (a) (b) Figure 2.20: The b2 = 2 binary trees on 2 vertices. The first few values of (2.3) are b0 = 1, b1 = 1, b2 = 2, b3 = 5, b4 = 14 which are rather small and of manageable size if we want to explicitly enumerate all different binary trees with the above orders. However, from n = 4 onwards the value of bn increases very fast. Instead of enumerating all the bn different binary trees of a specified order n, a related problem is generating a random binary tree of order n. That is, we consider the set B as a sample space of bn different binary trees on n vertices, and choose a random element from B. Such a random element can be generated using Algorithm 2.5. The list parent holds all vertices with less than two children, each vertex can be considered as a candidate parent to which we can add a child. An element of parent is a two-tuple (v, k) where the vertex v currently has k children. 2.4. Binary trees 79 (a) (b) (c) (d) (e) Figure 2.21: The b3 = 5 binary trees on 3 vertices. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) Figure 2.22: The b4 = 14 binary trees on 4 vertices. Algorithm 2.5 Random binary tree. Input Positive integer n. Output A random binary tree on n vertices. 1: if n = 1 then 2: return K1 3: v ← 0 4: T ← null graph 5: add v to T 6: parent ← [(v, 0)] 7: for i ← 1, 2, . . . , n − 1 do 8: (v, k) ← remove random element from parent 9: if k < 1 then 10: add (v, k + 1) to parent 11: add edge (v, i) to T 12: add (i, 0) to parent 13: return T 80 Chapter 2. Trees and forests 2.4.1 Binary codes What is a code? A code is a rule for converting data in one format, or well-defined tangible representation, into sequences of symbols in another format. The finite set of symbols used is called the alphabet. We shall identify a code as a finite set of symbols which are the image of the alphabet under this conversion rule. The elements of this set are referred to as codewords. For example, using the ASCII code, the letters in the English alphabet get converted into numbers in the set {0, 1, . . . , 255}. If these numbers are written in binary, then each codeword of a letter has length 8, i.e. eight bits. In this way, we can reformat or encode a “string” into a sequence of binary symbols, i.e. 0’s and 1’s. Encoding is the conversion process one way. Decoding is the reverse process, converting these sequences of code-symbols back into information in the original format. Codes are used for: Economy. Sometimes this is called entropy encoding since there is an entropy function which describes how much information a channel (with a given error rate) can carry and such codes are designed to maximize entropy as best as possible. In this case, in addition to simply being given an alphabet A, one might be given a weighted alphabet, i.e. an alphabet for which each symbol a ∈ A is associated with a nonnegative number wa ≥ 0 (in practice, this number represents the probability that the symbol a occurs in a typical word). Reliability. Such codes are called error-correcting codes, since such codes are de- signed to communicate information over a noisy channel in such a way that the errors in transmission are likely to be correctable. Security. Such codes are called cryptosystems. In this case, the inverse of the coding function c : A → B ∗ is designed to be computationally infeasible. In other words, the coding function c is designed to be a trapdoor function. Other codes are merely simpler ways to communicate information (e.g. flag semaphores, color codes, genetic codes, braille codes, musical scores, chess notation, football diagrams, and so on) and have little or no mathematical structure. We shall not study them. Basic definitions If every word in the code has the same length, the code is called a block code. If a code is not a block code, then it is called a variable-length code. A prefix-free code is a code (typically one of variable-length) with the property that there is no valid codeword in the code that is a prefix or start of any other codeword.1 This is the prefix-free condition. One example of a prefix-free code is the ASCII code. Another example is 00, 01, 100. On the other hand, a non-example is the code 00, 01, 010, 100 1 In other words, a codeword s = s1 · · · sm is a prefix of a codeword t = t1 · · · tn if and only if m ≤ n and s1 = t1 , . . . , sm = tm . Codes that are prefix-free are easier to decode than codes that are not prefix-free. 2.4. Binary trees 81 since the second codeword is a prefix of the third one. Another non-example is Morse code recalled in Table 2.1, where we use 0 for “·” (“dit”) and 1 for “−” (“dah”). For example, consider the Morse code for aand the Morse code for w. These codewords violate the prefix-free condition. A 01 N 10 B 1000 O 111 C 1010 P 0110 D 100 Q 1101 E 0 R 010 F 0010 S 000 G 110 T 1 H 0000 U 001 I 00 V 0001 J 0111 W 011 K 101 X 1001 L 0100 Y 1011 M 11 Z 1100 Table 2.1: Morse code Gray codes We begin with some history.2 Frank Gray (1887–1969) wrote about the so-called Gray codes in a 1951 paper published in the Bell System Technical Journal and then in 1953 patented a device (used for television sets) based on his paper. However, the idea of a binary Gray code appeared earlier. In fact, it appeared in an earlier patent (one by Stibitz in 1943). It was also used in the French engineer E. Baudot’s telegraph machine of 1878 and in a French booklet by L. Gros on the solution published in 1872 to the Chinese ring puzzle. The term “Gray code” is ambiguous. It is actually a large family of sequences of n-tuples. Let Zm = {0, 1, . . . , m − 1}. More precisely, an m-ary Gray code of length n (called a binary Gray code when m = 2) is a sequence of all possible (i.e. N = mn ) n-tuples g1 , g2 , . . . , gN where each gi ∈ Znm , gi and gi+1 differ by 1 in exactly one coordinate. In other words, an m-ary Gray code of length n is a particular way to order the set of all mn n-tuples whose coordinates are taken from Zm . From the transmission/commu- nication perspective, this sequence has two advantages: It is easy and fast to produce the sequence, since successive entries differ in only one coordinate. 2 This history comes from an unpublished section 7.2.1.1 (“Generating all n-tuples”) in volume 4 of Donald Knuth’s The Art of Computer Programming. 82 Chapter 2. Trees and forests An error is relatively easy to detect, since we can compare an n-tuple with the previous one. If they differ in more than one coordinate, we conclude that an error was made. Example 2.23. Here is a 3-ary Gray code of length 2: [0, 0], [1, 0], [2, 0], [2, 1], [1, 1], [0, 1], [0, 2], [1, 2], [2, 2] and the sequence [0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 1, 1], [1, 1, 1], [1, 0, 1], [0, 0, 1] is a binary Gray code of length 3. Gray codes have applications to engineering, recreational mathematics (solving the Tower of Hanoi puzzle, The Brain puzzle, the Chinese ring puzzle, etc.), and to math- ematics (e.g. aspects of combinatorics, computational group theory, and the computa- tional aspects of linear codes). Binary Gray codes Consider the so-called n-hypercube graph Qn , whose first few instances are illustrated in Figure 1.33. This can be envisioned as the graph whose vertices are the vertices of a cube in n-space {(x1 , . . . , xn ) | 0 ≤ xi ≤ 1} and whose edges are those line segments in Rn connecting two neighboring vertices, i.e. two vertices that differ in exactly one coordinate. A binary Gray code of length n can be regarded as a path on the hypercube graph Qn that visits each vertex of the cube exactly once. In other words, a binary Gray code of length n may be identified with a Hamiltonian path on the graph Qn . For example, Figure 2.23 illustrates a Hamiltonian path on Q3 . Figure 2.23: Viewing Γ3 as a Hamiltonian path on Q3 . How do we efficiently compute a Gray code? Perhaps the simplest way to state the idea of quickly constructing the reflected binary Gray code Γn of length n is as follows: Γ0 = [ ], Γn = [0, Γn−1 ], [1, Γrev n−1 ] 2.4. Binary trees 83 where Γrev m means the Gray code in reverse order. For instance, we have Γ0 = [ ], Γ1 = [0], [1] , Γ2 = [0, 0], [0, 1], [1, 1], [1, 0] and so on. This is a nice procedure for creating the entire list at once, which gets very long very fast. An implementation of the reflected Gray code using Python is given below. def graycode ( length , modulus ): """ Returns the n - tuple reflected Gray code mod m . EXAMPLES : sage : graycode (2 ,4) [[0 , 0] , [1 , 0] , [2 , 0] , [3 , 0] , [3 , 1] , [2 , 1] , [1 , 1] , [0 , 1] , [0 , 2] , [1 , 2] , [2 , 2] , [3 , 2] , [3 , 3] , [2 , 3] , [1 , 3] , [0 , 3]] """ n , m = length , modulus F = range ( m ) if n == 1: return [[ i ] for i in F ] L = graycode (n -1 , m ) M = [] for j in F : M = M +[ ll +[ j ] for ll in L ] k = len ( M ) Mr = [0]* m for i in range (m -1): i1 = i * int ( k / m ) # this requires Python 3.0 or Sage i2 = ( i +1)* int ( k / m ) Mr [ i ] = M [ i1 : i2 ] Mr [m -1] = M [( m -1)* int ( k / m ):] for i in range ( m ): if is_odd ( i ): Mr [ i ]. reverse () M0 = [] for i in range ( m ): M0 = M0 + Mr [ i ] return M0 Consider the reflected binary code of length 8, i.e. Γ8 . This has 28 = 256 codewords. Sage can easily create the list plot of the coordinates (x, y), where x is an integer j ∈ Z256 that indexes the codewords in Γ8 and the corresponding y is the j-th codeword in Γ8 converted to decimal. This will give us some idea of how the Gray code “looks” in some sense. The plot is given in Figure 2.24. What if we only want to compute the i-th Gray codeword in the Gray code of length n? Can it be computed quickly without computing the entire list? At least in the case of the reflected binary Gray code, there is a very simple way to do this. The k-th element in 84 Chapter 2. Trees and forests 200 100 0 0 50 100 150 200 250 Figure 2.24: Scatterplot of Γ8 . the above-described reflected binary Gray code of length n is obtained by simply adding the binary representation of k to the binary representation of the integer part of k/2. An example using Sage is given below. def int2binary (m , n ): ’’’ returns GF (2) vector of length n obtained from the binary repr of m , padded by 0 ’ s ( on the left ) to length n . EXAMPLES : sage : for j in range (8): ....: print int2binary (j ,3)+ int2binary ( int ( j /2) ,3) ....: (0 , 0 , 0) (0 , 0 , 1) (0 , 1 , 1) (0 , 1 , 0) (1 , 1 , 0) (1 , 1 , 1) (1 , 0 , 1) (1 , 0 , 0) ’’’ s = bin ( m ) k = len ( s ) F = GF (2) b = [ F (0)]* n for i in range (2 , k ): b [n - k + i ] = F ( int ( s [ i ])) return vector ( b ) def graycodeword (m , n ): ’’’ returns the k - th codeword in the reflected binary Gray code of length n . EXAMPLES : sage : graycodeword (3 ,3) (0 , 1 , 0) ’’’ return map ( int , int2binary (m , n )+ int2binary ( int ( m /2) , n )) 2.5 Huffman codes An alphabet A is a finite set whose elements are referred to as symbols. A word (or string or message) over A is a finite sequence of symbols in A and the length of the word is 2.5. Huffman codes 85 the number of symbols it contains. A word is usually written by concatenating symbols together, e.g. a1 a2 · · · ak (ai ∈ A) is a word of length k. A commonly occurring alphabet in practice is the binary alphabet B = {0, 1}. A word over the binary alphabet is a finite sequence of 0’s and 1’s. If A is an alphabet, let A∗ denote the set of all words in A. The length of a word is denoted by vertical bars. That is, if w = a1 · · · ak is a word over A, then define |w| : A∗ → Z by |w| = |a1 · · · ak | = k. Let A and B be two alphabets. A code for A using B is an injection c : A → B ∗ . By abuse of notation, we often denote the code simply by the set C = c(A) = {c(a) | a ∈ A}. The elements of C are called codewords. If B is the binary alphabet, then C is called a binary code. 2.5.1 Tree representation Any binary code can be represented by a tree, as Example 2.24 shows. Example 2.24. Let B` be the binary code of length ≤ `. Represent codewords of B` using trees. Solution. Here is how to represent the code B` consisting of all binary strings of length ≤ `. Start with the root node ε being the empty string. The two children of this node, v0 and v1 , correspond to the two strings of length 1. Label v0 with a “0” and v1 with a “1”. The two children of v0 , i.e. v00 and v01 , correspond to the strings of length 2 which start with a 0. Similarly, the two children of v1 , i.e. v10 and v11 , correspond to the strings of length 2 that each starts with a 1. Continue creating child nodes until we reach length `, at which point we stop. There are a total of 2`+1 − 1 nodes in this tree and 2` of them are leaves (vertices of a tree with degree 1, i.e. childless nodes). Note that the parent of any node is a prefix to that node. Label each node vs with the string “s”, where s is a binary sequence of length ≤ `. See Figure 2.25 for an example when ` = 2. ε 0 1 00 01 10 11 Figure 2.25: Tree representation of the binary code B2 . In general, if C is a code contained in B` , then to create the tree for C, start with the tree for B` . First, remove all nodes associated to a binary string for which it and 86 Chapter 2. Trees and forests all of its descendants are not in C. Next, remove all labels which do not correspond to codewords in C. The resulting labeled graph is the tree associated to the binary code C. For visualizing the construction of Huffman codes later, it is important to see that we can reverse this construction to start from such a binary tree and recover a binary code from it. The codewords are determined by the following rules: The root node gets the empty codeword. Each left-ward branch gets a 0 appended to the end of its parent. Each right-ward branch gets a 1 appended to the end. 2.5.2 Uniquely decodable codes If c : A → B∗ is a code, then we can extend c to A∗ by concatenation: c(a1 a2 · · · ak ) = c(a1 )c(a2 ) · · · c(ak ). If the extension c : A∗ → T ∗ is also an injection, then c is called uniquely decodable. The property of unique decodability or decipherability informally means that any given sequence of symbols has at most one interpretation as a sequence of codewords. Example 2.25. Is the Morse code in Table 2.1 uniquely decodable? Why or why not? Solution. Note that these Morse codewords all have lengths less than or equal to 4. Other commonly occurring symbols used (the digits 0 through 9, punctuation symbols, and some others) are also encodable in Morse code, but they use longer codewords. Let A denote the English alphabet, B = {0, 1} the binary alphabet, and c : A → B∗ the Morse code. Since c(ET ) = 01 = c(A), it is clear that the Morse code is not uniquely decodable. In fact, prefix-free implies uniquely decodable. Theorem 2.26. If a code c : A → B∗ is prefix-free, then it is uniquely decodable. Proof. We use induction on the length of a message. We want to show that if x1 · · · xk and y1 · · · y` are messages with c(x1 ) · · · c(xk ) = c(y1 ) · · · c(y` ), then x1 · · · xk = y1 · · · y` . This in turn implies k = ` and xi = yi for all i. The case of length 1 follows from the fact that c : A → B ∗ is injective (by the definition of code). Suppose that the statement of the theorem holds for all codes of length < m. We must show that the length m case is true. Suppose c(x1 ) · · · c(xk ) = c(y1 ) · · · c(y` ), where m = max(k, `). These strings are equal, so the substring c(x1 ) of the left-hand side and the substring c(y1 ) of the right-hand side are either equal or one is contained in the other. If, say, c(x1 ) is properly contained in c(y1 ), then c is not prefix-free. Likewise if c(y1 ) is properly contained in c(x1 ). Therefore, c(x1 ) = c(y1 ), which implies x1 = y1 . Now remove this codeword from both sides, so c(x2 ) · · · c(xk ) = c(y2 ) · · · c(y` ). By the induction hypothesis, x2 · · · xk = y2 · · · y` . These facts together imply k = ` and xi = yi for all i. 2.5. Huffman codes 87 P Consider now a weighted alphabet (A, p), where p : A → [0, 1] satisfies a∈A p(a) = 1, and a code c : A → B∗ . In other words, p is a probability distribution on A. Think of p(a) as the probability that the symbol a arises in a typical message. The average word length L(c) is3 X L(c) = p(a) · |c(a)| a∈A where | · | is the length of a codeword. Given a weighted alphabet (A, p) as above, a code c : A → B∗ is called optimal if there is no such code with a smaller average word length. Optimal codes satisfy the following amazing property. For a proof, which is very easy and highly recommended for anyone who is curious to see more, refer to section 3.6 of Biggs [28]. Lemma 2.27. Suppose c : A → B∗ is a binary optimal prefix-free code and let ` = maxa∈A |c(a)| denote the maximum length of a codeword. The following statements hold. 1. If |c(a0 )| > |c(a)|, then p(a0 ) ≤ p(a). 2. The subset of codewords of length `, i.e. C` = {c ∈ c(A) | ` = |c(a)|} contains two codewords of the form b0 and b1 for some b ∈ B∗ . 2.5.3 Huffman coding The Huffman code construction is based on the second property in Lemma 2.27. Using this property, in 1952 David Huffman [104] presented an optimal prefix-free binary code, which has since been named Huffman code. Here is the recursive/inductive construction of a Huffman code. We shall regard the binary Huffman code as a tree, as described above. Suppose that the weighted alphabet (A, p) has n symbols. We assume inductively that there is an optimal prefix-free binary code for any weighted alphabet (A0 , p0 ) having < n symbols. Huffman’s rule 1 Let a, a0 ∈ A be symbols with the smallest weights. Construct a new weighted alphabet with a, a0 replaced by the single symbol a∗ = aa0 and having weight p(a∗ ) = p(a) + p(a0 ). All other symbols and weights remain unchanged. Huffman’s rule 2 For the code (A0 , p0 ) above, if a∗ is encoded as the binary string s, then the encoded binary string for a is s0 and the encoded binary string for a0 is s1. The above two rules tell us how to inductively build the tree representation for the Huffman code of (A, p) up from its leaves (associated to the low weight symbols). Find two different symbols of lowest weight, a and a0 . If two such symbols do not exist, stop. Replace the weighted alphabet with the new weighted alphabet as in Huffman’s rule 1. 3 In probability terminology, this is the expected value E(X) of the random variable X, which assigns to a randomly selected symbol in A the length of the associated codeword in c. 88 Chapter 2. Trees and forests Add two nodes (labeled with a and a0 , respectively) to the tree, with parent a∗ (see Huffman’s rule 1). If there are no remaining symbols in A, label the parent a∗ with the empty set and stop. Otherwise, go to the first step. These ideas are captured in Algorithm 2.6, which outlines steps to construct a binary tree corresponding to the Huffman code of an alphabet. Line 2 initializes a minimum- priority queue Q with the symbols in the alphabet A. Line 3 creates an empty binary tree that will be used to represent the Huffman code corresponding to A. The for loop from lines 4 to 10 repeatedly extracts from Q two elements a and b of minimum weights. We then create a new vertex z for the tree T and also let a and b be vertices of T . The weight W [z] of z is the sum of the weights of a and b. We let z be the parent of a and b, and insert the new edges za and zb into T . The newly created vertex z is now inserted into Q with priority W [z]. After n − 1 rounds of the for loop, the priority queue has only one element in it, namely the root r of the binary tree T . We extract r from Q (line 11) and return it together with T (line 12). Algorithm 2.6 Binary tree representation of Huffman codes. Input An alphabet A of n symbols. A weight list W of size n such that W [i] is the weight of ai ∈ A. Output A binary tree T representing the Huffman code of A and the root r of T . 1: n ← |A| 2: Q ← A . minimum priority queue 3: T ← empty tree 4: for i ← 1, 2, . . . , n − 1 do 5: a ← extractMin(Q) 6: b ← extractMin(Q) 7: z ← node with left child a and right child b 8: add the edges za and zb to T 9: W [z] ← W [a] + W [b] 10: insert z into priority queue Q 11: r ← extractMin(Q) 12: return (T, r) The runtime analysis of Algorithm 2.6 depends on the implementation of the priority queue Q. Suppose Q is a simple unsorted list. The initialization on line 2 requires O(n) time. The for loop from line 4 to 10 is executed exactly n − 1 times. Searching Q to determine the element of minimum weight requires time at most O(n). Determining two elements of minimum weights requires time O(2n). The for loop requires time O(2n2 ), which is also the time requirement for the algorithm. An efficient implementation of the priority queue Q, e.g. as a binary minimum heap, can lower the running time of Algorithm 2.6 down to O(n log2 (n)). Algorithm 2.6 represents the Huffman code of an alphabet as a binary tree T rooted at r. For an illustration of the process of constructing a Huffman tree, see Figure 2.26. To determine the actual encoding of each symbol in the alphabet, we feed T and r to Algorithm 2.7 to obtain the encoding of each symbol. Starting from the root r whose designated label is the empty string ε, the algorithm traverses the vertices of T in a 2.5. Huffman codes 89 13 2 2 19 4 5 2 7 3 13 2 1 19 4 5 2 7 1 3 1 1 (a) (b) 13 4 19 4 5 2 7 3 13 4 19 4 5 5 7 2 2 2 2 2 3 1 1 1 1 (c) (d) 13 8 19 5 5 7 13 8 19 10 7 4 4 2 3 4 4 5 5 2 2 2 2 2 3 1 1 1 1 (e) (f) 13 15 19 10 23 15 19 7 8 5 5 10 13 7 8 4 4 2 3 5 5 4 4 2 2 2 3 2 2 1 1 1 1 (g) (h) 57 23 34 23 34 10 13 15 19 10 13 15 19 5 5 7 8 5 5 7 8 2 3 4 4 2 3 4 4 2 2 2 2 1 1 1 1 (i) (j) Figure 2.26: Constructing a Huffman tree. 90 Chapter 2. Trees and forests breadth-first search fashion. If v is an internal vertex with label e, the label of its left- child is the concatenation e0 and for the right-child of v we assign the label e1. If v happens to be a leaf vertex, we take its label to be its Huffman encoding. Any Huffman encoding assigned to a symbol of an alphabet is not unique. Either of the two children of an internal vertex can be designated as the left- (respectively, right-) child. The runtime of Algorithm 2.7 is O(|V |), where V is the vertex set of T . Algorithm 2.7 Huffman encoding of an alphabet. Input A binary tree T representing the Huffman code of an alphabet A. The root r of T. Output A list H representing a Huffman code of A, where H[ai ] corresponds to a Huffman encoding of ai ∈ A. 1: H ← [ ] . list of Huffman encodings 2: Q ← [r] . queue of vertices 3: while length(Q) > 0 do 4: root ← dequeue(Q) 5: if root is a leaf then 6: H[root] ← label of root 7: else 8: a ← left child of root 9: b ← right child of root 10: enqueue(Q, a) 11: enqueue(Q, b) 12: label of a ← label of root + 0 13: label of b ← label of root + 1 14: return H Example 2.28. Consider the alphabet A = {a, b, c, d, e, f } with corresponding weights w(a) = 19, w(b) = 2, w(c) = 40, w(d) = 25, w(e) = 31, and w(f ) = 3. Construct a binary tree representation of the Huffman code of A and determine the encoding of each symbol of A. Solution. Use Algorithm 2.6 to construct a binary tree representation of the weighted alphabet A. The resulting binary tree T is shown in Figure 2.27(a), where ai : wi is an abbreviation for “vertex ai has weight wi ”. The binary tree is rooted at k. To encode each alphabetic symbol, input T and k into Algorithm 2.7 to get the encodings shown in Figure 2.27(b). 2.6 Tree traversals In computer science, tree traversal refers to the process of examining each vertex in a tree data structure. Starting at the root of an ordered tree T , we can traverse the vertices of T in one of various ways. A level-order traversal of an ordered tree T examines the vertices in increasing order of depth, with vertices of equal depth being examined according to their prescribed order. One way to think about level-order traversal is to consider vertices of T having the same depth as being ordered from left to right in decreasing order of importance. 2.6. Tree traversals 91 k : 120 ε i : 49 j : 71 0 1 h : 24 d : 25 e : 31 c : 40 00 01 10 11 g:5 a : 19 000 001 b:2 f :3 0000 0001 (a) (b) Figure 2.27: Binary tree representation of an alphabet and its Huffman encodings. If [v1 , v2 , . . . , vn ] lists the vertices from left to right at depth k, a decreasing order of importance can be realized by assigning each vertex a numeric label using a labelling function L : V (T ) → R such that L(v1 ) < L(v2 ) < · · · < L(vn ). In this way, a vertex with a lower numeric label is examined prior to a vertex with a higher numeric label. A level-order traversal of T , whose vertices of equal depth are prioritized according to L, is an examination of the vertices of T from top to bottom, left to right. As an example, the level-order traversal of the tree in Figure 2.28 is 42, 4, 15, 2, 3, 5, 7, 10, 11, 12, 13, 14. Our discussion is formalized in Algorithm 2.8, whose general structure mimics that of breadth-first search. For this reason, level-order traversal is also known as breadth-first traversal. Each vertex is enqueued and dequeued exactly once. The while loop is executed n times, hence we have a runtime of O(n). Another name for level-order traversal is top- down traversal because we first visit the root node and then work our way down the tree, increasing the depth as we move downward. Algorithm 2.8 Level-order traversal. Input An ordered tree T on n > 0 vertices. Output A list of the vertices of T in level-order. 1: L ← [ ] 2: Q ← empty queue 3: r ← root of T 4: enqueue(Q, r) 5: while length(Q) > 0 do 6: v ← dequeue(Q) 7: append(L, v) 8: [u1 , u2 , . . . , uk ] ← ordering of children of v 9: for i ← 1, 2, . . . , k do 10: enqueue(Q, ui ) 11: return L 92 Chapter 2. Trees and forests 42 4 15 2 3 5 7 10 11 12 13 14 Figure 2.28: Traversing a tree. Pre-order traversal is a traversal of an ordered tree using a general strategy similar to depth-first search. For this reason, pre-order traversal is also referred to as depth-first traversal. Parents are visited prior to their respective children and siblings are visited according to their prescribed order. The pseudocode for pre-order traversal is presented in Algorithm 2.9. Note the close resemblance to Algorithm 2.8; the only significant change is to use a stack instead of a queue. Each vertex is pushed and popped exactly once, so the while loop is executed n times, resulting in a runtime of O(n). Using Algorithm 2.9, a pre-order traversal of the tree in Figure 2.28 is 42, 4, 2, 3, 10, 11, 14, 5, 12, 13, 15, 7. Algorithm 2.9 Pre-order traversal. Input An ordered tree T on n > 0 vertices. Output A list of the vertices of T in pre-order. 1: L ← [ ] 2: S ← empty stack 3: r ← root of T 4: push(S, r) 5: while length(S) > 0 do 6: v ← pop(S) 7: append(L, v) 8: [u1 , u2 , . . . , uk ] ← ordering of children of v 9: for i ← k, k − 1, . . . , 1 do 10: push(S, ui ) 11: return L Whereas pre-order traversal lists a vertex v the first time we visit it, post-order traversal lists v the last time we visit it. In other words, children are visited prior to their respective parents, with siblings being visited in their prescribed order. The prefix “pre” in “pre-order traversal” means “before”, i.e. visit parents before visiting children. On the other hand, the prefix “post” in “post-order traversal” means “after”, i.e. visit parents after having visited their children. The pseudocode for post-order traversal is presented 2.6. Tree traversals 93 in Algorithm 2.10, whose general structure bears close resemblance to Algorithm 2.9. The while loop of the former is executed n times because each vertex is pushed and popped exactly once, resulting in a runtime of O(n). The post-order traversal of the tree in Figure 2.28 is 2, 10, 14, 11, 3, 12, 13, 5, 4, 7, 15, 42. Algorithm 2.10 Post-order traversal. Input An ordered tree T on n > 0 vertices. Output A list of the vertices of T in post-order. 1: L ← [ ] 2: S ← empty stack 3: r ← root of T 4: push(S, r) 5: while length(S) > 0 do 6: if top(S) is unmarked then 7: mark top(S) 8: [u1 , u2 , . . . , uk ] ← ordering of children of top(S) 9: for i ← k, k − 1, . . . , 1 do 10: push(S, ui ) 11: else 12: v ← pop(S) 13: append(L, v) 14: return L Instead of traversing a tree T from top to bottom as is the case with level-order traversal, we can reverse the direction of our traversal by traversing a tree from bottom to top. Called bottom-up traversal , we first visit all the leaves of T and consider the subtree T1 obtained by vertex deletion of those leaves. We then recursively perform bottom-up traversal of T1 by visiting all of its leaves and obtain the subtree T2 resulting from vertex deletion of those leaves of T1 . Apply bottom-up traversal to T2 and its vertex deletion subtrees until we have visited all vertices, including the root vertex. The result is a procedure for bottom-up traversal as presented in Algorithm 2.11. In lines 3 to 5, we initialize the list C to contain the number of children of vertex i. This takes O(m) time, where m = |E(T )|. Lines 6 to 14 extract all the leaves of T and add them to the queue Q. From lines 15 to 23, we repeatedly apply bottom-up traversal to subtrees of T . As each vertex is enqueued and dequeued exactly once, the two loops together run in time O(n) and therefore Algorithm 2.11 has a runtime of O(n + m). As an example, a bottom-up traversal of the tree in Figure 2.28 is 2, 7, 10, 12, 13, 14, 15, 5, 11, 3, 4, 42. Yet another common tree traversal technique is called in-order traversal . However, in- order traversal is only applicable to binary trees, whereas the other traversal techniques we considered above can be applied to any tree with at least one vertex. Given a binary tree T having at least one vertex, in-order traversal first visits the root of T and consider its left- and right-children. We then recursively apply in-order traversal to the left and right subtrees of the root vertex. Notice the symmetry in our description of in-order traversal: start at the root, then traverse the left and right subtrees in in-order. For this 94 Chapter 2. Trees and forests Algorithm 2.11 Bottom-up traversal. Input An ordered tree T on n > 0 vertices. Output A list of the vertices of T in bottom-up order. 1: Q ← empty queue 2: r ← root of T 3: C ← [0, 0, . . . , 0] . n copies of 0 4: for each edge (u, v) ∈ E(T ) do 5: C[u] ← C[u] + 1 6: R ← empty queue 7: enqueue(R, r) 8: while length(R) > 0 do 9: v ← dequeue(R) 10: for each w ∈ children(v) do 11: if C[w] = 0 then 12: enqueue(Q, w) 13: else 14: enqueue(R, w) 15: L ← [ ] 16: while length(Q) > 0 do 17: v ← dequeue(Q) 18: append(L, v) 19: if v 6= r then 20: C[parent(v)] ← C[parent(v)] − 1 21: if C[parent(v)] = 0 then 22: u ← parent(v) 23: enqueue(Q, u) 24: return L Algorithm 2.12 In-order traversal. Input A binary tree T on n > 0 vertices. Output A list of the vertices of T in in-order. 1: L ← [ ] 2: S ← empty stack 3: v ← root of T 4: while true do 5: if v 6= Null then 6: push(S, v) 7: v ← left-child of v 8: else 9: if length(S) = 0 then 10: exit the loop 11: v ← pop(S) 12: append(L, v) 13: v ← right-child of v 14: return L 2.7. Problems 95 reason, in-order traversal is sometimes referred to as symmetric traversal. Our discussion is summarized in Algorithm 2.12. In the latter algorithm, if a vertex does not have a left-child, then the operation of finding its left-child returns NULL. The same holds when the vertex does not have a right-child. Since each vertex is pushed and popped exactly once, it follows that in-order traversal runs in time O(n). Using Algorithm 2.12, an in-order traversal of the tree in Figure 2.27(b) is 0000, 000, 0001, 00, 001, 0, 01, ε, 10, 1, 11. 2.7 Problems When solving problems, dig at the roots instead of just hacking at the leaves. — Anthony J. D’Angelo, The College Blue Book 2.1. Construct all nonisomorphic trees of order 7. 2.2. Let G be a weighted connected graph and let T be a subgraph of G. Then T is a maximum spanning tree of G provided that the following conditions are satisfied: (a) T is a spanning tree of G. (b) The total weight of T is maximum among all spanning trees of G. Modify Kruskal’s, Prim’s, and Borůvka’s algorithms to return a maximum spanning tree of G. 2.3. Describe and present pseudocode of an algorithm to construct all spanning trees of a connected graph. What is the worst-case runtime of your algorithm? How many of the constructed spanning trees are nonisomorphic to each other? Repeat the exercise for minimum and maximum spanning trees. 2.4. Consider an undirected, connected simple graph G = (V, E) of order n and size m and having an integer weight function w : E → Z given by w(e) > 0 for all e ∈ E. Suppose that G has N minimum spanning trees. Yamada et al. [198] provide an O(N m ln n) algorithm to construct all the N minimum spanning trees of G. Describe and provide pseudocode of the Yamada-Kataoka-Watanabe algorithm. Provide runtime analysis and prove the correctness of this algorithm. 2.5. The solution of Example 2.3 relied on the following result: Let T = (V, E) be a tree rooted at v0 and suppose v0 has exactly two children. If maxv∈V deg(v) = 3 and v0 is the only vertex with degree 2, then T is a binary tree. Prove this statement. Give examples of graphs that are binary trees but do not satisfy the conditions of the result. Under which conditions would the above test return an incorrect answer? 2.6. What is the worst-case runtime of Algorithm 2.1? 2.7. Figure 2.5 shows two nonisomorphic spanning trees of the 4 × 4 grid graph. (a) For each n = 1, 2, . . . , 7, construct all nonisomorphic spanning trees of the n × n grid graph. 96 Chapter 2. Trees and forests (b) Explain and provide pseudocode of an algorithm for constructing all spanning trees of the n × n grid graph, where n > 0. (c) In general, if n is a positive integer, how many nonisomorphic spanning trees are there in the n × n grid graph? (d) Describe and provide pseudocode of an algorithm to generate a random span- ning tree of the n × n grid graph. What is the worst-case runtime of your algorithm? 2.8. Theorem 2.4 shows how to recursively construct a new tree from a given collection of trees, hence it can be considered as a recursive definition of trees. To prove the- orems based upon recursive definitions, we use a proof technique called structural induction. Let S(C) be a statement about the collection of structures C, each of which is defined by a recursive definition. In the base case, prove S(C) for the basis structure(s) C. For the inductive case, let X be a structure formed using the recursive definition from the structures Y1 , Y2 , . . . , Yk . Assume for induction that the statements S(Y1 ), S(Y2 ), . . . , S(Yk ) hold and use the inductive hypotheses S(Yi ) to prove S(X). Hence conclude that S(X) is true for all X. Apply structural induction to show that any graph constructed using Theorem 2.4 is indeed a tree. 2.9. In Kruskal’s Algorithm 2.2, line 5 requires that the addition of a new edge to T does not result in T having a cycle. A tree by definition has no cycles. Suppose line 5 is changed to: if ei ∈ / E(T ) and T ∪ {ei } is a tree then With this change, explain why Algorithm 2.2 would return a minimum spanning tree or why the algorithm would fail to do so. 2.10. This problem is concerned with improving the runtime of Kruskal’s Algorithm 2.2. Explain how to use a priority queue to obviate the need for sorting the edges by weight. Investigate the union-find data structure. Explain how to use union-find to ensure that the addition of each edge results in an acyclic graph. 2.11. Figure 2.29 shows a weighted version of the Chvátal graph, which has 12 ver- tices and 24 edges. Use this graph as input to Kruskal’s, Prim’s, and Borůvka’s algorithms and compare the resulting minimum spanning trees. 2.12. Algorithm 2.1 presents a randomized procedure to construct a spanning tree of a given connected graph via repeated edge deletion. (a) Describe and present pseudocode of a randomized algorithm to grow a span- ning tree via edge addition. (b) Would Algorithm 2.1 still work if the input graph G has self-loops or multiple edges? Explain why or why not. If not, modify Algorithm 2.1 to handle the case where G has self-loops and multiple edges. (c) Repeat the previous exercise for Kruskal’s, Prim’s, and Borůvka’s algorithms. 2.13. Algorithm 2.13 constructs a random spanning tree of the complete graph Kn on n > 0 vertices. Its runtime is dependent on efficient algorithms for obtaining a random permutation of a set of objects, and choosing a random element from a given set. 2.7. Problems 97 0 11.4 40.7 17.1 14.4 5.6 35.4 5 1 4 14.5 15 6 9 11.8 8.5 48 3.7 11 10 0.2 9.1 27.1 17 6.9 43.2 10.2 42.7 7 22.1 8 22 36.6 2 3 44.2 Figure 2.29: Weighted Chvátal graph. Algorithm 2.13 Random spanning tree of Kn . Input A positive integer n representing the order of Kn , with vertex set V = {0, 1, . . . , n − 1}. Output A random spanning tree of Kn . 1: if n = 1 then 2: return K1 3: P ← random permutation of V 4: T ← null tree 5: for i ← 1, 2, . . . , n − 1 do 6: j ← random element from {0, 1, . . . , i − 1} 7: add edge (P [j], P [i]) to T 8: return T 98 Chapter 2. Trees and forests (a) Describe and analyze the runtime of a procedure to construct a random per- mutation of a set of nonnegative integers. (b) Describe an algorithm for randomly choosing an element of a set of nonnega- tive integers. Analyze the runtime of this algorithm. (c) Taking into consideration the previous two algorithms, what is the runtime of Algorithm 2.13? 2.14. We want to generate a random undirected, connected simple graph on n vertices and having m edges. Start by generating a random spanning tree T of Kn . Then add random edges to T until the requirements are satisfied. (a) Present pseudocode to realize the above procedure. What is the worst-case runtime of your algorithm? (b) Modify your algorithm to handle the case where m < n − 1. Why must m ≥ n − 1? (c) Modify your algorithm to handle the case where each edge has a weight within the closed interval [α, β]. 2.15. Enumerate all the different binary trees on 5 vertices. 2.16. Algorithm 2.5 generates a random binary tree on n > 0 vertices. Modify this algorithm so that it generates a random k-ary tree of order n > 0, where k ≥ 3. 2.17. Show by giving an example that the Morse code is not prefix-free. 2.18. Consider the alphabet A = {a, b, c} with corresponding probabilities (or weights) p(a) = 0.5, p(b) = 0.3, and p(c) = 0.2. Generate two different Huffman codes for A and illustrate the tree representations of those codes. 2.19. Find the Huffman code for the letters of the English alphabet weighted by the frequency of common American usage.4 2.20. Let G = (V1 , E2 ) be a graph and T = (V2 , E2 ) a spanning tree of G. Show that there is a one-to-one correspondence between fundamental cycles in G and edges not in T . 2.21. Let G = (V, E) be the 3 × 3 grid graph and T1 = (V1 , E1 ), T2 = (V2 , E2 ) be spanning trees of G in Example 2.1. Find a fundamental cycle in G for T1 that is not a fundamental cycle in G for T2 . 2.22. Usually there exist many spanning trees of a graph. Classify those graphs for which there is only one spanning tree. In other words, find necessary and sufficient conditions for a graph G such that if T is a spanning tree of G then T is unique. 2.23. Convert the function graycodeword into a pure Python function. 2.24. Example 2.13 verifies that for any positive integer n > 1, repeated iteration of the Euler phi function ϕ(n) eventually produces 1. Show that this is the case or provide an explanation why it is in general false. 4 You can find this on the Internet or in the literature. Part of this exercise is finding this frequency distribution yourself. 2.7. Problems 99 2.25. The Collatz conjecture [126] asserts that for any integer n > 0, repeated iteration of the function ( 3n+1 2 , if n is odd, T (n) = n 2 , if n is even eventually produces the value 1. For example, repeated iteration of T (n) starting from n = 22 results in the sequence 22, 11, 17, 26, 13, 20, 10, 5, 8, 4, 2, 1. (2.4) One way to think about the Collatz conjecture is to consider the digraph G produced by considering (ai , T (ai )) as a directed edge of G. Then the Collatz conjecture can be rephrased to say that there is some integer k > 0 such that (ak , T (ak )) = (2, 1) is a directed edge of G. The graph obtained in this man- ner is called the Collatz graph of T (n). Given a collection of positive integers α1 , α2 , . . . , αk , let Gαi be the Collatz graph of the function T (αi ) with initial iter- ation value αi . Then the union of the Gαi is the directed tree [ Gαi i rooted at 1, called the Collatz tree of (α1 , α2 , . . . , αk ). Figure 2.30 shows such a tree for the collection of initial iteration values 1024, 336, 340, 320, 106, 104, and 96. See Lagarias [127, 128] for a comprehensive survey of the Collatz conjecture. (a) The Collatz sequence of a positive integer n > 1 is the integer sequence pro- duced by repeated iteration of T (n) with initial iteration value n. For example, the Collatz sequence of n = 22 is the sequence (2.4). Write a Sage function to produce the Collatz sequence of an integer n > 1. (b) The Collatz length of n > 1 is the number of terms in the Collatz sequence of n, inclusive of the starting iteration value and the final integer 1. For instance, the Collatz length of 22 is 12, that of 106 is 11, and that of 51 is 18. Write a Sage function to compute the Collatz length of a positive integer n > 1. If n > 1 is a vertex in a Collatz tree, verify that the Collatz length of n is the distance d(n, 1). (c) Describe the Collatz graph produced by the function T (n) with initial iteration value n = 1. (d) Fix a positive integer n > 1 and let Li be the Collatz length of the integer 1 ≤ i ≤ n. Plot the pairs (i, Li ) on one set of axes. 2.26. The following result was first published in Wiener [196]. Let T = (V, E) be a tree of order n > 0. For each edge e ∈ E, let n1 (e) and n2 (e) = n − n1 (e) be the orders of the two components of the edge-deletion subgraph T − e. Show that the Wiener number of T is X W (T ) = n1 (e) · n2 (e). e∈E 2.27. The following result [147] was independently discovered in the late 1980s by Merris and McKay, and is known as the Merris-McKay theorem. Let T be a tree of order 100 Chapter 2. Trees and forests 1 2 4 8 16 5 32 3 10 21 64 6 20 42 128 12 13 40 84 85 256 24 26 80 168 170 512 48 52 53 160 336 340 1024 96 104 106 320 Figure 2.30: The union of Collatz graphs is a tree. 2.7. Problems 101 n and let L be its Laplacian matrix having eigenvalues λ1 , λ2 , . . . , λn . Show that the Wiener number of T is n−1 X 1 W (T ) = n . i=1 λ i 2.28. For each of the algorithms below: (i) justify whether or not it can be applied to multigraphs or multidigraphs; (ii) if not, modify the algorithm so that it is applicable to multigraphs or multidigraphs. (a) Randomized spanning tree construction Algorithm 2.1. (b) Kruskal’s Algorithm 2.2. (c) Prim’s Algorithm 2.3. (d) Borůvka’s Algorithm 2.4. 2.29. Section 2.6 provides iterative algorithms for the following tree traversal techniques: (a) Level-order traversal: Algorithm 2.8. (b) Pre-order traversal: Algorithm 2.9. (c) Post-order traversal: Algorithm 2.10. (d) Bottom-up traversal: Algorithm 2.11. (e) In-order traversal: Algorithm 2.12. Rewrite each of the above as recursive algorithms. 2.30. In cryptography, the Merkle signature scheme [145] was introduced in 1987 as an alternative to traditional digital signature schemes such as the Digital Signature Algorithm or RSA. Buchmann et al. [43] and Szydlo [178] provide efficient algo- rithms for speeding up the Merkle signature scheme. Investigate this scheme and how it uses binary trees to generate digital signatures. 2.31. Consider the finite alphabet A = {a1 , a2 , . . . , ar }. If C is a subset of A∗ , then we say that C is an r-ary code and call r the radix of the code. McMillan’s theorem [143], first published in 1956, relates codeword lengths to unique decipherability. In particular, let C = {c1 , c2 , . . . , cn } be an r-ary code where each ci has length `i . If C is uniquely decipherable, McMillan’s theorem states that the codeword lengths `i must satisfy Kraft’s inequality Xn 1 ≤ 1. i=1 r`i Prove McMillan’s theorem. 2.32. A code C = {c1 , c2 , . . . , cn } is said to be instantaneous if each codeword ci can be interpreted as soon as it is received. For example, given the the code {01, 010} and the string 01010, upon receiving the first 0 we are unable to decide whether that element belong to 01 or 010. However, the code {1, 01} is instantaneous because given the string 1101 and the first 1, we can interpret the latter as the codeword 1. Prove that a code is instantaneous if and only if it is prefix-free. 102 Chapter 2. Trees and forests 2.33. Kraft’s inequality and the accompanying Kraft’s theorem were first published [124] in 1949 in the Master’s thesis of Leon Gordon Kraft. Kraft’s theorem relates the inequality to instantaneous codes. Let C = {c1 , c2 , . . . , cn } be an r-ary code where each codeword ci has length `i . Kraft’s theorem states that C is an instantaneous code if and only if the codeword lengths satisfy Xn 1 `i ≤ 1. i=1 r Prove Kraft’s theorem. 2.34. Let T be a nontrivial tree and P let ni count the number of vertices of T that have degree i. Show that T has 2 + ∞ i=3 (i − 2)ni leaves. 2.35. If a forest F has k trees totalling n vertices altogether, how many edges does F contain? 2.36. The Lucas number Ln , named after Édouard Lucas, has the following recursive definition: 2, if n = 0, Ln = 1, if n = 1, Ln−1 + Ln−2 , if n > 1. √ (a) If ϕ = (1 + 5)/2 is the golden ratio, show that Ln = ϕn + (−ϕ)−n . (b) Let τ (Wn ) be the number of spanning trees of the wheel graph. Benjamin and Yerger [22] provide a combinatorial proof that τ (Wn ) = L2n − 2. Present the Benjamin-Yerger combinatorial proof. (c) Let G be the Dodecahedral graph, implemented in Sage as G = graphs.DodecahedralGraph( Does its cutset matrix satisfy the undirected analog of Theorem 2.19? Chapter 3 Shortest paths algorithms — Randall Munroe, xkcd, http://xkcd.com/518/ Graph algorithms have many applications. Suppose you are a salesman with a product you would like to sell in several cities. To determine the cheapest travel route from city- to-city, you must effectively search a graph having weighted edges for the “cheapest” route visiting each city once. Each vertex denotes a city you must visit and each edge has a weight indicating either the distance from one city to another or the cost to travel from one city to another. Shortest path algorithms are some of the most important algorithms in algorithmic graph theory. In this chapter, we first examine several common graph traversal algo- rithms and some basic data structures underlying these algorithms. A data structure is a combination of methods for structuring a collection of data (e.g. vertices and edges) and protocols for accessing the data. We then consider a number of common shortest path algorithms, which rely in one way or another on graph traversal techniques and basic data structures for organizing and managing vertices and edges. 103 104 Chapter 3. Shortest paths algorithms 3.1 Representing graphs in a computer To err is human but to really foul things up requires a computer. — Anonymous, Farmers’ Almanac for 1978, “Capsules of Wisdom” In section 1.3, we discussed how to use matrices for representing graphs and digraphs. If A = [aij ] is an m×n matrix, the adjacency matrix representation of a graph would require representing all the mn entries of A. Alternative graph representations exist that are much more efficient than representing all entries of a matrix. The graph representation used can be influenced by the size of a graph or the purpose of the representation. Sec- tion 3.1.1 discusses the adjacency list representation that can result in less storage space requirement than the adjacency matrix representation. The graph6 format discussed in section 3.1.3 provides a compact means of storing graphs for archival purposes. 3.1.1 Adjacency lists A list is a sequence of objects. Unlike sets, a list may contain multiple copies of the same object. Each object in a list is referred to as an element of the list. A list L of n ≥ 0 elements is written as L = [a1 , a2 , . . . , an ], where the i-th element ai can be indexed as L[i]. In case n = 0, the list L = [ ] is referred to as the empty list. Two lists are equivalent if they both contain the same elements at exactly the same positions. Define the adjacency lists of a graph as follows. Let G be a graph with vertex set V = {v1 , v2 , . . . , vn }. Assign to each vertex vi a list Li containing all the vertices that are adjacent to vi . The list Li associated with vi is referred to as the adjacency list of vi . Then Li = [ ] if and only if vi is an isolated vertex. We say that Li is the adjacency list of vi because any permutation of the elements of Li results in a list that contains the same vertices adjacent to vi . We are mainly concerned with the neighbors of vi , but disregard the position where each neighbor is located in Li . If each adjacency list Li contains si elements where 0 ≤ si ≤ n, we say that Li has length P si . The adjacency list representation of the graph G requires that we represent i si = 2 · |E(G)| ≤ n2 elements in a computer’s memory, since each edge appears twice in the adjacency list representation. An adjacency list is explicit about which vertices are adjacent to a vertex and implicit about which vertices are not adjacent to that same vertex. Without knowing the graph G, given the adjacency lists L1 , L2 , . . . , Ln , we can reconstruct G. For example, Figure 3.1 shows a graph and its adjacency list representation. 4 3 7 8 6 L1 = [2, 8] L5 = [6, 8] L2 = [1, 6] L6 = [2, 5, 8] 5 L3 = [4] L7 = [ ] 1 2 L4 = [3] L8 = [1, 5, 6] Figure 3.1: A graph and its adjacency lists. Example 3.1. The Kneser graph with parameters (n, k), also known as the (n, k)-Kneser graph, is the graph whose vertices are all the k-subsets of {1, 2, . . . , n}. Furthermore, two 3.1. Representing graphs in a computer 105 vertices are adjacent if their corresponding sets are disjoint. Draw the (5, 2)-Kneser graph and find its order and adjacency lists. In general, if n and k are positive, what is the order of the (n, k)-Kneser graph? Solution. The (5, 2)-Kneser graph is the graph whose vertices are the 2-subsets {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5} of {1, 2, 3, 4, 5}. That is, each vertex of the (5, 2)-Kneser graph is5×4 a 2-combination of the 5 set {1, 2, 3, 4, 5} and therefore the graph itself has order 2 = 2! = 10. The edges of this graph are ({1, 3}, {2, 4}), ({2, 4}, {1, 5}), ({2, 4}, {3, 5}), ({1, 3}, {4, 5}), ({1, 3}, {2, 5}) ({3, 5}, {1, 4}), ({3, 5}, {1, 2}), ({1, 4}, {2, 3}), ({1, 4}, {2, 5}), ({4, 5}, {2, 3}) ({4, 5}, {1, 2}), ({1, 5}, {2, 3}), ({1, 5}, {3, 4}), ({3, 4}, {1, 2}), ({3, 4}, {2, 5}) from which we obtain the following adjacency lists: L{1,2} = [{3, 4}, {3, 5}, {4, 5}], L{1,3} = [{2, 4}, {2, 5}, {4, 5}], L{1,4} = [{2, 3}, {3, 5}, {2, 5}], L{1,5} = [{2, 4}, {3, 4}, {2, 3}], L{2,3} = [{1, 5}, {1, 4}, {4, 5}], L{2,4} = [{1, 3}, {1, 5}, {3, 5}], L{2,5} = [{1, 3}, {3, 4}, {1, 4}], L{3,4} = [{1, 2}, {1, 5}, {2, 5}], L{3,5} = [{2, 4}, {1, 2}, {1, 4}], L{4,5} = [{1, 3}, {1, 2}, {2, 3}]. The (5, 2)-Kneser graph itself is shown in Figure 3.2. Using Sage, we have sage : K = graphs . KneserGraph (5 , 2); K Kneser graph with parameters 5 ,2: Graph on 10 vertices sage : for v in K . vertices (): ... print (v , K . neighbors ( v )) ... ({4 , 5} , [{1 , 3} , {1 , 2} , {2 , 3}]) ({1 , 3} , [{2 , 4} , {2 , 5} , {4 , 5}]) ({2 , 5} , [{1 , 3} , {3 , 4} , {1 , 4}]) ({2 , 3} , [{1 , 5} , {1 , 4} , {4 , 5}]) ({3 , 4} , [{1 , 2} , {1 , 5} , {2 , 5}]) ({3 , 5} , [{2 , 4} , {1 , 2} , {1 , 4}]) ({1 , 4} , [{2 , 3} , {3 , 5} , {2 , 5}]) ({1 , 5} , [{2 , 4} , {3 , 4} , {2 , 3}]) ({1 , 2} , [{3 , 4} , {3 , 5} , {4 , 5}]) ({2 , 4} , [{1 , 3} , {1 , 5} , {3 , 5}]) If n and k are positive integers, then the (n, k)-Kneser graph has n n(n − 1) · · · (n − k + 1) = k k! vertices. We can categorize a graph G = (V, E) as dense or sparse based upon its size. A dense 2 2 graph has size |E| that is close to |V | , i.e. |E| = Ω |V | , in which case it is feasible to 2 represent G as an adjacency matrix. The size of a sparse graph is much less than |V | , i.e. |E| = Ω |V | , which renders the adjacency matrix representation as unsuitable. For a sparse graph, an adjacency list representation can require less storage space than an adjacency matrix representation of the same graph. 106 Chapter 3. Shortest paths algorithms {3, 4} {1, 2} {1, 5} {2, 5} {2, 3} {3, 5} {4, 5} {1, 4} {2, 4} {1, 3} Figure 3.2: The (5, 2)-Kneser graph. 3.1.2 Edge lists Lists can also be used to store the edges of a graph. To create an edge list L for a graph G, if uv is an edge of G then we let uv or the ordered pair (u, v) be an element of L. In general, let v0 v1 , v2 v3 , . . . , vk vk+1 be all the edges of G, where k is even. Then the edge list of G is given by L = [v0 v1 , v2 v3 , . . . , vk vk+1 ]. In some cases, it is desirable to have the edges of G be in contiguous list representation. If the edge list L of G is as given above, the contiguous edge list representation of the edges of G is [v0 , v1 , v2 , v3 , . . . , vk , vk+1 ]. That is, if 0 ≤ i ≤ k is even then vi vi+1 is an edge of G. 3.1.3 The graph6 format The graph formats graph6 and sparse6 were developed by Brendan McKay [141] at The Australian National University as a compact way to represent graphs. These two formats use bit vectors and printable characters of the American Standard Code for Information Interchange (ASCII) encoding scheme. The 64 printable ASCII characters used in graph6 and sparse6 are those ASCII characters with decimal codes from 63 to 126, inclusive, as shown in Table 3.1. This section shall only cover the graph6 format. For full specification on both of the graph6 and sparse6 formats, see McKay [141]. Bit vectors Before discussing how graph6 and sparse6 represent graphs using printable ASCII char- acters, we first present encoding schemes used by these two formats. A bit vector is, as 3.1. Representing graphs in a computer 107 binary decimal glyph binary decimal glyph 0111111 63 ? 1011111 95 _ 1000000 64 @ 1100000 96 ‘ 1000001 65 A 1100001 97 a 1000010 66 B 1100010 98 b 1000011 67 C 1100011 99 c 1000100 68 D 1100100 100 d 1000101 69 E 1100101 101 e 1000110 70 F 1100110 102 f 1000111 71 G 1100111 103 g 1001000 72 H 1101000 104 h 1001001 73 I 1101001 105 i 1001010 74 J 1101010 106 j 1001011 75 K 1101011 107 k 1001100 76 L 1101100 108 l 1001101 77 M 1101101 109 m 1001110 78 N 1101110 110 n 1001111 79 O 1101111 111 o 1010000 80 P 1110000 112 p 1010001 81 Q 1110001 113 q 1010010 82 R 1110010 114 r 1010011 83 S 1110011 115 s 1010100 84 T 1110100 116 t 1010101 85 U 1110101 117 u 1010110 86 V 1110110 118 v 1010111 87 W 1110111 119 w 1011000 88 X 1111000 120 x 1011001 89 Y 1111001 121 y 1011010 90 Z 1111010 122 z 1011011 91 [ 1111011 123 { 1011100 92 \ 1111100 124 | 1011101 93 ] 1111101 125 } 1011110 94 ^ 1111110 126 ~ Table 3.1: ASCII printable characters used by graph6 and sparse6. 108 Chapter 3. Shortest paths algorithms its name suggests, a vector whose elements are 1’s and 0’s. It can be represented as a list of bits, e.g. E can be represented as the ASCII bit vector [1, 0, 0, 0, 1, 0, 1]. For brevity, we write a bit vector in a compact form such as 1000101. The length of a bit vector is its number of bits. The most significant bit of a bit vector v is the bit position with the largest value among all the bit positions in v. Similarly, the least significant bit is the bit position in v having the least value among all the bit positions in v. The least significant bit of v is usually called the parity bit because when v is interpreted as an integer the parity bit determines whether the integer is even or odd. Reading 1000101 from left to right, the first bit 1 is the most significant bit, followed by the second bit 0 which is the second most significant bit, and so on all the way down to the seventh bit 1 which is the least significant bit. The order in which we process the bits of a bit vector v = bn−1 bn−2 · · · b0 (3.1) is referred to as endianness. Processing v in big-endian order means that we first process the most significant bit of v, followed by the second most significant bit, and so on all the way down to the least significant bit of v. Thus, in big-endian order we read the bits bi of v from left to right in increasing order of powers of 2. Table 3.2 illustrates the big-endian interpretation of the ASCII binary representation of E. Little-endian order means that we first process the least significant bit, followed by the second least significant bit, and so on all the way up to the most significant bit. In little-endian order, the bits bi are read from right to left in increasing order of powers of 2. Table 3.3 illustrates the little-endian interpretation of the ASCII binary representation of E. In his novel Gulliver’s Travels first published in 1726, Jonathan Swift used the terms big- and little-endian in satirizing politicians who squabbled over whether to break an egg at the big end or the little end. Danny Cohen [55, 56] first used the terms in 1980 as an April fool’s joke in the context of computer architecture. Suppose the bit vector (3.1) is read in big-endian order. To determine the integer representation of v, multiply each bit value by its corresponding position value, then add up all the results. Thus, if v is read in big-endian order, the integer representation of v is obtained by evaluating the polynomial n−1 X p(x) = xi bi = xn−1 bn−1 + xn−2 bn−2 + · · · + xb1 + b0 . (3.2) i=0 at x = 2. See problem 3.?? for discussion of an efficient method to compute the integer representation of a bit vector. position 0 1 2 3 4 5 6 bit value 1 0 0 0 1 0 1 position value 20 21 22 23 24 25 26 Table 3.2: Big-endian order of the ASCII binary code of E. In graph6 and sparse6 formats, the length of a bit vector must be a multiple of 6. Suppose v is a bit vector of length k such that 6 - k. To transform v into a bit vector having length a multiple of 6, let r = k mod 6 be the remainder upon dividing k by 6, and pad 6 − r zeros to the right of v. 3.1. Representing graphs in a computer 109 position 0 1 2 3 4 5 6 bit value 1 0 0 0 1 0 1 position value 26 25 24 23 22 21 20 Table 3.3: Little-endian order of the ASCII binary code of E. Suppose v = b1 b2 · · · bk is a bit vector of length k, where 6 | k. We split v into k/6 bit vectors vi , each of length 6. For 0 ≤ i ≤ k/6, the i-th bit vector is given by vi = b6i−5 b6i−4 b6i−3 b6i−2 b6i−1 b6i . Consider each vi as the big-endian binary representation of a positive integer. Use (3.2) to obtain the integer representation Ni of each vi . Then add 63 to each Ni to obtain Ni0 and store Ni0 in one byte of memory. That is, each Ni0 can be represented as a bit vector of length 8. Thus the required number of bytes to store v is dk/6e. Let Bi be the byte representation of Ni0 so that R(v) = B1 B2 · · · Bdk/6e (3.3) denotes the representation of v as a sequence of dk/6e bytes. We now discuss how to encode an integer n in the range 0 ≤ n ≤ 236 − 1 using (3.3) and denote such an encoding of n as N (n). Let v be the big-endian binary representation of n. Then N (n) is given by n + 63, if 0 ≤ n ≤ 62, N (n) = 126 R(v), if 63 ≤ n ≤ 258047, (3.4) 126 126 R(v), if 258048 ≤ n ≤ 236 − 1. Note that n + 63 requires one byte of storage memory, while 126 R(v) and 126 126 R(v) require 4 and 8 bytes, respectively. The graph6 format The graph6 format is used to represent simple, undirected graphs of order from 0 to 236 − 1, inclusive. Let G be a simple, undirected graph of order 0 ≤ n ≤ 236 − 1. If n = 0, then G is represented in graph6 format as “?”. Suppose n > 0. Let M = [aij ] be the adjacency matrix of G. Consider the upper triangle of M , excluding the main diagonal, and write that upper triangle as the bit vector v = a0,1 a0,2 a1,2 a0,3 a1,3 a2,3 · · · a0,i a1,i · · · ai−1,i · · · a0,n a1,n · · · an−1,n |{z} | {z } | {z } | {z } | {z } c1 c2 c3 ci cn where ci denotes the entries a0,i a1,i · · · ai−1,i in column i of M . Then the graph6 repre- sentation of G is N (n)R(v), where R(v) and N (n) are as in (3.3) and (3.4), respectively. That is, N (n) encodes the order of G and R(v) encodes the edges of G. 110 Chapter 3. Shortest paths algorithms 3.2 Graph searching Errors, like straws, upon the surface flow; He who would search for pearls must dive below. — John Dryden, All for Love, 1678 This section discusses two fundamental algorithms for graph traversal: breadth-first search and depth-first search. The word “search” used in describing these two algorithms is rather misleading. It would be more accurate to describe them as algorithms for constructing trees using the adjacency information of a given graph. However, the names “breadth-first search” and “depth-first search” are entrenched in literature on graph theory and computer science. From hereon, we use these two names as given above, bearing in mind their intended purposes. 3.2.1 Breadth-first search Breadth-first search (BFS) is a strategy for running through the vertices of a graph. It was presented by Moore [148] in 1959 within the context of traversing mazes. Lee [131] independently discovered the same algorithm in 1961 in his work on routing wires on circuit boards. In the physics literature, BFS is also known as a “burning algorithm” in view of the analogy of a fire burning and spreading through an area, a piece of paper, fabric, etc. The basic BFS algorithm can be described as follows. Starting from a given vertex v of a graph G, we first explore the neighborhood of v by visiting all vertices that are adjacent to v. We then apply the same strategy to each of the neighbors of v. The strategy of exploring the neighborhood of a vertex is applied to all vertices of G. The result is a tree rooted at v and this tree is a subgraph of G. Algorithm 3.1 presents a general template for the BFS strategy. The tree resulting from the BFS algorithm is called a breadth-first search tree. Algorithm 3.1 A general breadth-first search template. Input A directed or undirected graph G = (V, E) of order n > 0. A vertex s from which to start the search. The vertices are numbered from 1 to n = |V |, i.e. V = {1, 2, . . . , n}. Output A list D of distances of all vertices from s. A tree T rooted at s. 1: Q ← [s] . queue of nodes to visit 2: D ← [∞, ∞, . . . , ∞] . n copies of ∞ 3: D[s] ← 0 4: T ← [ ] 5: while length(Q) > 0 do 6: v ← dequeue(Q) 7: for each w ∈ adj(v) do 8: if D[w] = ∞ then 9: D[w] ← D[v] + 1 10: enqueue(Q, w) 11: append(T, vw) 12: return (D, T ) 3.2. Graph searching 111 The breadth-first search algorithm makes use of a special type of list called a queue. This is analogous to a queue of people waiting in line to be served. A person may enter the queue by joining the rear of the queue. The person who is in the queue the longest amount of time is served first, followed by the person who has waited the second longest time, and so on. Formally, a queue Q is a list of elements. At any time, we only have access to the first element of Q, known as the front or start of the queue. We insert a new element into Q by appending the new element to the rear or end of the queue. The operation of removing the front of Q is referred to as dequeue, while the operation of appending to the rear of Q is called enqueue. That is, a queue implements a first-in first-out (FIFO) protocol for adding and removing elements. As with lists, the length of a queue is its total number of elements. 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (a) Original undirected graph. (b) First iteration of while loop. 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (c) Second iteration of while loop. (d) Third iteration of while loop. 1 1 2 2 5 3 7 3 7 6 4 6 5 4 (e) Fourth iteration of while loop. (f) Final BFS tree. Figure 3.3: Breadth-first search tree for an undirected graph. Note that the BFS Algorithm 3.1 works on both undirected and directed graphs. For an undirected graph, line 7 means that we explore all the neighbors of vertex v, i.e. the set adj(v) of vertices adjacent to v. In the case of a digraph, we replace “w ∈ adj(v)” on line 7 with “w ∈ oadj(v)” because we only want to explore all vertices that are out- neighbors of v. The algorithm returns two lists D and T . The list T contains a subset 112 Chapter 3. Shortest paths algorithms 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (a) Original digraph. (b) First iteration of while loop. 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (c) Second iteration of while loop. (d) Third iteration of while loop. 4 1 2 3 6 3 7 2 5 7 4 6 5 1 (e) Fourth iteration of while loop. (f) Final BFS tree. Figure 3.4: Breadth-first search tree for a digraph. 3.2. Graph searching 113 of edges in E(G) that make up a tree rooted at the given start vertex s. As trees are connected graphs without cycles, we may take the vertices comprising the edges of T to be the vertex set of the tree. It is clear that T represents a tree by means of a list of edges, which allows us to identify the tree under consideration as the edge list T . The list D has the same number of elements as the order of G = (V, E), i.e. length(D) = |V |. The i-th element D[i] counts the number of edges in T between the vertices s and vi . In other words, D[i] is the length of the s-vi path in T . It can be shown that D[i] = ∞ if and only if G is disconnected. After one application of Algorithm 3.1, it may happen that D[i] = ∞ for at least one vertex vi ∈ V . To traverse those vertices that are unreachable from s, again we apply Algorithm 3.1 on G with starting vertex vi . Repeat this algorithm as often as necessary until all vertices of G are visited. The result may be a tree that contains all the vertices of G or a collection of trees, each of which contains a subset of V (G). Figures 3.3 and 3.4 present BFS trees resulting from applying Algorithm 3.1 on an undirected graph and a digraph, respectively. Theorem 3.2. The worst-case time complexity of Algorithm 3.1 is O(|V | + |E|). Proof. Without loss of generality, we can assume that G = (V, E) is connected. The initialization steps in lines 1 to 4 take O(|V |) time. After initialization, all but one vertex are labelled ∞. Line 8 ensures that each vertex is enqueued at most once and hence dequeued at most once. Each of enqueuing and dequeuing takes constant time. The total time devoted to queue operations is O(|V |). The adjacency list of a vertex is scanned after dequeuing that vertex, so each adjacency list is scanned at most once. Summing the lengths of the adjacency lists, we have Θ(|E|) and therefore we require O(|E|) time to scan the adjacency lists. After the adjacency list of a vertex is scanned, at most k edges are added to the list T , where k is the length of the adjacency list under consideration. Like queue operations, appending to a list takes constant time, hence we require O(|E|) time to build the list T . Therefore, BFS runs in O(|V | + |E|) time. Theorem 3.3. For the list D resulting from Algorithm 3.1, let s be a starting vertex and let v be a vertex such that D[v] 6= ∞. Then D[v] is the length of any shortest path from s to v. Proof. It is clear that D[v] = ∞ if and only if there are no paths from s to v. Let v be a vertex such that D[v] 6= ∞. As v can be reached from s by a path of length D[v], the length d(s, v) of any shortest s-v path satisfies d(s, v) ≤ D[v]. Use induction on d(s, v) to show that equality holds. For the base case s = v, we have d(s, v) = D[v] = 0 since the trivial path has length zero. Assume for induction that if d(s, v) = k, then d(s, v) = D[v]. Let d(s, u) = k + 1 with the corresponding shortest s-u path being (s, v1 , v2 , . . . , vk , u). By our induction hypothesis, (s, v1 , v2 , . . . , vk ) is a shortest path from s to vk of length d(s, vk ) = D[vk ] = k. In other words, D[vk ] < D[u] and the while loop spanning lines 5 to 11 processes vk before processing u. The graph under consideration has the edge vk u. When examining the adjacency list of vk , BFS reaches u (if u is not reached earlier) and so D[u] ≤ k + 1. Hence, D[u] = k + 1 and therefore d(s, u) = D[u] = k + 1. In the proof of Theorem 3.3, we used d(u, v) to denote the length of the shortest path from u to v. This shortest path length is also known as the distance from u to v, and will be discussed in further details in section 3.3 and Chapter 5. The diameter diam(G) of a graph G = (V, E) is defined as diam(G) = max d(u, v). (3.5) u,v∈V u6=v 114 Chapter 3. Shortest paths algorithms Using the above definition, to find the diameter we first determine the distance between each pair of distinct vertices, then we compute the maximum of all such distances. Breadth-first search is a useful technique for finding the diameter: we simply run breadth- first search from each vertex. An interesting application of the diameter appears in the small-world phenomenon [120, 146, 193], which contends that a certain special class of sparse graphs have low diameter. 3.2.2 Depth-first search — Randall Munroe, xkcd, http://xkcd.com/761/ A depth-first search (DFS) is a graph traversal strategy similar to breadth-first search. Both BFS and DFS differ in how they explore each vertex. Whereas BFS explores the neighborhood of a vertex v before moving on to explore the neighborhoods of the neighbors, DFS explores as deep as possible a path starting at v. One can think of BFS as exploring the immediate surrounding, while DFS prefers to see what is on the other side of the hill. In the 19th century, Lucas [138] and Tarry [182] investigated DFS as a strategy for traversing mazes. Fundamental properties of DFS were discovered in the early 1970s by Hopcroft and Tarjan [101, 181]. To get an intuitive appreciation for DFS, suppose we have an 8 × 8 chessboard in front of us. We place a single knight piece on a fixed square of the board, as shown in Figure 3.5(a). Our objective is to find a sequence of knight moves that visits each and every square exactly once, while obeying the rules of chess that govern the movement of the knight piece. Such a sequence of moves, if one exists, is called a knight’s tour . How do we find such a tour? We could make one knight move after another, recording each move to ensure that we do not step on a square that is already visited, until we could not make any more moves. Acknowledging defeat when encountering a dead end, 3.2. Graph searching 115 0Z0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0Z0 Z0Z0Z0Z0 0Z0Z0Z0Z 0Z0Z0Z0Z Z0Z0m0Z0 Z0Z0Z0Z0 0Z0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0Z0 Z0Z0Z0Z0 0Z0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0Z0 Z0Z0Z0Z0 (a) The knight’s initial position. (b) A knight’s tour. (c) Graph representation of the tour. Figure 3.5: The knight’s tour from a given starting position. 116 Chapter 3. Shortest paths algorithms it might make sense to backtrack a few moves and try again, hoping we would not get stuck. If we fail again, we try backtracking a few more moves and traverse yet another path, hoping to make further progress. Repeat this strategy until a tour is found or until we have exhausted all possible moves. The above strategy for finding a knight’s tour is an example of depth-first search, sometimes called backtracking. Figure 3.5(b) shows a knight’s tour with the starting position as shown in Figure 3.5(a); and Figure 3.5(c) is a graph representation of this tour. The black-filled nodes indicate the endpoints of the tour. A more interesting question is: What is the number of knight’s tours on an 8 × 8 chessboard? Loebbing and Wegener [136] announced in 1996 that this number is 33,439,123,484,294. The answer was later corrected by McKay [142] to be 13,267,364,410,532. See [69] for a discussion of the knight’s tour and its relationship to mathematics. Algorithm 3.2 A general depth-first search template. Input A directed or undirected graph G = (V, E) of order n > 0. A vertex s from which to start the search. The vertices are numbered from 1 to n = |V |, i.e. V = {1, 2, . . . , n}. Output A list D of distances of all vertices from s. A tree T rooted at s. 1: S ← [s] . stack of nodes to visit 2: D ← [∞, ∞, . . . , ∞] . n copies of ∞ 3: D[s] ← 0 4: T ← [ ] 5: while length(S) > 0 do 6: v ← pop(S) 7: for each w ∈ adj(v) do 8: if D[w] = ∞ then 9: D[w] ← D[v] + 1 10: push(S, w) 11: append(T, vw) 12: return (D, T ) Algorithm 3.2 formalizes the above description of depth-first search. The tree re- sulting from applying DFS on a graph is called a depth-first search tree. The general structure of this algorithm bears close resemblance to Algorithm 3.1. A significant dif- ference is that instead of using a queue to structure and organize vertices to be visited, DFS uses another special type of list called a stack . To understand how elements of a stack are organized, we use the analogy of a stack of cards. A new card is added to the stack by placing it on top of the stack. Any time we want to remove a card, we are only allowed to remove the top-most card that is on the top of the stack. A list L = [a1 , a2 , . . . , ak ] of k elements is a stack when we impose the same rules for element insertion and removal. The top and bottom of the stack are L[k] and L[1], respectively. The operation of removing the top element of the stack is referred to as popping the element off the stack. Inserting an element into the stack is called pushing the element onto the stack. In other words, a stack implements a last-in first-out (LIFO) protocol for element insertion and removal, in contrast to the FIFO policy of a queue. We also use the term length to refer to the number of elements in the stack. The depth-first search Algorithm 3.2 can be analyzed similar to how we analyzed Algorithm 3.3. Just as BFS is applicable to both directed and undirected graphs, we 3.2. Graph searching 117 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (a) Original undirected graph. (b) First iteration of while loop. 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (c) Second iteration of while loop. (d) Third iteration of while loop. 1 2 5 3 6 7 4 (e) Final DFS tree. Figure 3.6: Depth-first search tree for an undirected graph. 118 Chapter 3. Shortest paths algorithms 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (a) Original digraph. (b) First iteration of while loop. 1 2 1 2 3 3 7 7 4 4 6 5 6 5 (c) Second iteration of while loop. (d) Third iteration of while loop. 4 1 2 3 6 3 7 5 7 4 6 5 1 2 (e) Fourth iteration of while loop. (f) Final DFS tree. Figure 3.7: Depth-first search tree for a digraph. 3.2. Graph searching 119 can also have undirected graphs and digraphs as input to DFS. For the case of an undirected graph, line 7 of Algorithm 3.2 considers all vertices adjacent to the current vertex v. In case the input graph is directed, we replace “w ∈ adj(v)” on line 7 with “w ∈ oadj(v)” to signify that we only want to consider the out-neighbors of v. If any neighbors (respectively, out-neighbors) of v are labelled as ∞, we know that we have not explored any paths starting from any of those vertices. So we label each of those unexplored vertices with a positive integer and push them onto the stack S, where they will wait for later processing. We also record the paths leading from v to each of those unvisited neighbors, i.e. the edges vw for each vertex w ∈ adj(v) (respectively, w ∈ oadj(v)) are appended to the list T . The test on line 8 ensures that we do not push onto S any vertices on the path that lead to v. When we resume another round of the while loop that starts on line 5, the previous vertex v have been popped off S and the neighbors (respectively, out-neighbors) of v have been pushed onto S. For example, in step 2 of Figure 3.6, vertex 5 is considered in DFS (in contrast to the vertex 2 in step 2 of the BFS in the graph in Figure 3.3) because DFS is organized by the LIFO protocol (in contrast to the FIFO protocol of BFS). To explore a path starting at v, we choose any unexplored neighbors of v by popping an element off S and repeat the for loop starting on line 7. Repeat the DFS algorithm as often as required in order to traverse all vertices of the input graph. The output of DFS consists of two lists D and T : T is a tree rooted at the starting vertex s; and each D[i] counts the length of the s-vi path in T . Figures 3.6 and 3.7 show the DFS trees resulting from running Algorithm 3.2 on an undirected graph and a digraph, respectively. The worst-case time complexity of DFS can be analyzed using an argument similar to that in Theorem 3.2. Arguing along the same lines as in the proof of Theorem 3.3, we can also show that the list D returned by DFS contains lengths of any shortest paths in the tree T from the starting vertex s to any other vertex in T (but not necessarily for shortest paths in the original graph G). 0 5 1 4 6 9 7 8 2 3 Figure 3.8: The Petersen graph. Example 3.4. In 1898, Julius Petersen published [161] a graph that now bears his name: the Petersen graph shown in Figure 3.8. Compare the search trees resulting from running breadth- and depth-first searches on the Petersen graph with starting vertex 0. Solution. The Petersen graph in Figure 3.8 can be constructed and searched as follows. sage : g = graphs . PetersenGraph (); g Petersen graph : Graph on 10 vertices 120 Chapter 3. Shortest paths algorithms sage : list ( g . breadth_first_search (0)) [0 , 1 , 4 , 5 , 2 , 6 , 3 , 9 , 7 , 8] sage : list ( g . depth_first_search (0)) [0 , 5 , 8 , 6 , 9 , 7 , 2 , 3 , 4 , 1] From the above Sage session, we see that starting from vertex 0 breadth-first search yields the edge list [01, 04, 05, 12, 16, 43, 49, 57, 58] and depth-first search produces the corresponding edge list [05, 58, 86, 69, 97, 72, 23, 34, 01]. Our results are illustrated in Figure 3.9. 0 0 5 5 1 4 1 4 6 9 6 9 7 8 7 8 2 3 2 3 (a) Breadth-first search. (b) Depth-first search. Figure 3.9: Traversing the Petersen graph starting from vertex 0. 3.2.3 Connectivity of a graph Both BFS and DFS can be used to determine if an undirected graph is connected. Let G = (V, E) be an undirected graph of order n > 0 and let s be an arbitrary vertex of G. We initialize a counter c ← 1 to mean that we are starting our exploration at s, hence we have already visited one vertex, i.e. s. We apply either BFS or DFS, treating G and s as input to any of these algorithms. Each time we visit a vertex that was previously unvisited, we increment the counter c. At the end of the algorithm, we compare c with n. If c = n, we know that we have visited all vertices of G and conclude that G is connected. Otherwise, we conclude that G is disconnected. This procedure is summarized in Algorithm 3.3. Note that Algorithm 3.3 uses the BFS template of Algorithm 3.1, with some minor changes. Instead of initializing the list D with n = |V | copies of ∞, we use n copies of 0. Each time we have visited a vertex w, we make the assignment D[w] ← 1, instead of incrementing the value D[v] of w’s parent vertex and Passign that value to D[w]. At the end of the while loop, we have the equality c = d∈D d. The value of this sum could be used in the test starting from line 12. However, the value of the counter c is incremented immediately after we have visited an unvisited vertex. An advantage is that we do not need to perform a separate summation outside of the while loop. To use the DFS template for determining graph connectivity, we simply replace the queue implementation in Algorithm 3.3 with a stack implementation (see problem 3.14). 3.3. Weights and distances 121 Algorithm 3.3 Determining whether an undirected graph is connected. Input An undirected graph G = (V, E) of order n > 0. A vertex s from which to start the search. The vertices are numbered from 1 to n = |V |, i.e. V = {1, 2, . . . , n}. Output true if G is connected; false otherwise. 1: Q ← [s] . queue of nodes to visit 2: D ← [0, 0, . . . , 0] . n copies of 0 3: D[s] ← 1 4: c ← 1 5: while length(Q) > 0 do 6: v ← dequeue(Q) 7: for each w ∈ adj(v) do 8: if D[w] = 0 then 9: D[w] ← 1 10: c←c+1 11: enqueue(Q, w) 12: if c = |V | then 13: return true 14: return false 3.3 Weights and distances In Chapter 1, we briefly mentioned some applications of weighted graphs, but we did not define the concept of weighted graphs. A graph is said to be weighted when we assign a numeric label or weight to each of its edges. Depending on the application, we can let the vertices represent physical locations and interpret the weight of an edge as the distance separating two adjacent vertices. There might be a cost involved in traveling from a vertex to one of its neighbors, in which case the weight assigned to the corresponding edge can represent such a cost. The concept of weighted digraphs can be similarly defined. When no explicit weights are assigned to the edges of an undirected graph or digraph, it is usually convenient to consider each edge as having a weight of one or unit weight. Based on the concept of weighted graphs, we now define what it means for a path to be a shortest path. Let G = (V, E) be a (di)graph with nonnegative edge weights w(e) ∈ R for each edge e ∈ E. The length or distance d(P ) of a u-v path P from u ∈ V to v ∈ V is the sum of the edge weights for edges in P . Denote by d(u, v) the smallest value of d(P ) for all paths P from u to v. When we regard edge weights as physical distances, a u-v path that realizes d(u, v) is sometimes called a shortest path from u to v. The above definitions of distance and shortest path also apply to graphs with negative edge weights. Unless otherwise specified, where the weight of an edge is not explicitly given, we usually consider the edge to have unit weight. The distance function d on a graph with nonnegative edge weights is known as a metric function. Intuitively, the distance between two physical locations is greater than zero. When these two locations coincide, i.e. they are one and the same location, the distance separating them is zero. Regardless of whether we are measuring the distance from location a to b or from b to a, we would obtain the same distance. Imagine now a third location c. The distance from a to b plus the distance from b to c is greater than or equal to the distance from a to c. The latter principle is known as the triangle 122 Chapter 3. Shortest paths algorithms inequality. In summary, given three vertices u, v, w in a graph G, the distance function d on G satisfies the following property. Lemma 3.5. Path distance as metric function. Let G = (V, E) be a graph with weight function w : E → R. Define a distance function d : V × V → R given by ( ∞, if there are no paths from u to v, d(u, v) = min{w(W ) | W is a u-v walk}, otherwise. Then d is a metric on V if it satisfies the following properties: 1. Nonnegativity: d(u, v) ≥ 0 with d(u, v) = 0 if and only if u = v. 2. Symmetry: d(u, v) = d(v, u). 3. Triangle inequality: d(u, v) + d(v, w) ≥ d(u, w). The pair (V, d) is called a metric space, where the word “metric” refers to the distance function d. Any graphs we consider are assumed to have finite sets of vertices. For this reason, (V, d) is also known as a finite metric space. The distance matrix D = [d(vi , vj )] of a connected graph is the distance matrix of its finite metric space. The topic of metric space is covered in further details in topology texts such as Runde [168] and Shirali and Vasudeva [173]. See Buckley and Harary [44] for an in-depth coverage of the distance concept in graph theory. Many different algorithms exist for computing a shortest path in a weighted graph. Some only work if the graph has no negative weight cycles. Some assume that there is a single start or source vertex. Some compute the shortest paths from any vertex to any other and also detect if the graph has a negative weight cycle. No matter what algorithm is used for the special case of nonnegative weights, the length of the shortest path can neither equal nor exceed the order of the graph. Lemma 3.6. Fix a vertex v in a connected graph G = (V, E) of order n = |V |. If there are no negative weight cycles in G, then there exists a shortest path from v to any other vertex w ∈ V that uses at most n − 1 edges. Proof. Suppose that G contains no negative weight cycles. Observe that at most n − 1 edges are required to construct a path from v to any vertex w (Proposition 1.13). Let P denote such a path: P : v0 = v, v1 , v2 , . . . , vk = w. Since G has no negative weight cycles, the weight of P is no less than the weight of P 0 , where P 0 is the same as P except that all cycles have been removed. Thus, we can remove all cycles from P and obtain a v-w path P 0 of lower weight. Since the final path is acyclic, it must have no more than n − 1 edges. Having defined weights and distances, we are now ready to discuss shortest path algorithms for weighted graphs. The breadth-first search Algorithm 3.1 can be applied where each edge has unit weight. Moving on to the general case of graphs with positive edge weights, algorithms for determining shortest paths in such graphs can be classified as weight-setting or weight-correcting [83]. A weight-setting method traverses a graph and assigns weights that, once assigned, remain unchanged for the duration of the al- gorithm. Weight-setting algorithms cannot deal with negative weights. On the other 3.3. Weights and distances 123 Algorithm 3.4 A template for shortest path algorithms. Input A weighted graph or digraph G = (V, E), where the vertices are numbered as V = {1, 2, . . . , n}. A starting vertex s. Output A list D of distances from s to all other vertices. A list P of parent vertices such that P [v] is the parent of v. 1: D ← [∞, ∞, . . . , ∞] . n copies of ∞ 2: C ← list of candidate vertices to visit 3: while length(C) > 0 do 4: select v ∈ C 5: C ← remove(C, v) 6: for each u ∈ adj(v) do 7: if D[u] > D[v] + w(vu) then 8: D[u] ← D[v] + w(vu) 9: P [u] ← v 10: if u ∈ / C then 11: add u to C 12: return (D, P ) hand, a weight-correcting method is able to change the value of a weight many times while traversing a graph. In contrast to a weight-setting algorithm, a weight-correcting algorithm is able to deal with negative weights, provided that the weight sum of any cycle is nonnegative. The term negative cycle refers to the weight sum s of a cycle such that s < 0. Some algorithms halt upon detecting a negative cycle; examples of such algorithms include the Bellman-Ford and Johnson’s algorithms. Algorithm 3.4 is a general template for many shortest path algorithms. With a tweak here and there, one could modify it to suit the problem at hand. Note that w(vu) is the weight of the edge vu. If the input graph is undirected, line 6 considers all the neighbors of v. For digraphs, we are interested in out-neighbors of v and accordingly we replace “u ∈ adj(v)” in line 6 with “u ∈ oadj(v)”. The general flow of Algorithm 3.4 follows the same pattern as depth-first and breadth-first searches. 124 Chapter 3. Shortest paths algorithms 3.4 Dijkstra’s algorithm — Randall Munroe, xkcd, http://xkcd.com/342/ Dijkstra’s algorithm [61], discovered by E. W. Dijkstra in 1959, is a graph search algo- rithm that solves the single-source shortest path problem for a graph with nonnegative edge weights. The algorithm is a generalization of breadth-first search. Imagine that the vertices of a weighted graph represent cities and edge weights represent distances between pairs of cities connected by a direct road. Dijkstra’s algorithm can be used to find a shortest route from a fixed city to any other city. Let G = (V, E) be a (di)graph with nonnegative edge weights. Fix a start or source vertex s ∈ V . Dijkstra’s Algorithm 3.5 performs a number of steps, basically one step for each vertex in V . First, we initialize a list D with n copies of ∞ and then assign 0 to D[s]. The purpose of the symbol ∞ is to denote the largest possible value. The list D is to store the distances of all shortest paths from s to any other vertices in G, where we take the distance of s to itself to be zero. The list P of parent vertices is initially empty and the queue Q is initialized to all vertices in G. We now consider each vertex in Q, removing any vertex after we have visited it. The while loop starting on line 5 runs until we have visited all vertices. Line 6 chooses which vertex to visit, preferring a vertex v whose distance value D[v] from s is minimal. After we have determined such a vertex v, we remove it from the queue Q to signify that we have visited v. The for loop starting on line 8 adjusts the distance values of each neighbor u of v such that u is also in Q. If G is directed, we only consider out-neighbors of v that are also in Q. The conditional starting on line 9 is where the adjustment takes place. The expression D[v] + w(vu) sums the distance from s to v and the distance from v to u. If this total sum is less than the distance D[u] from s to u, we assign this lesser distance to D[u] and let v be the parent vertex of u. In this way, we are choosing a neighbor vertex that results in minimal distance from s. Each pass through the while loop decreases the number of elements in Q by one without adding any elements to Q. Eventually, we would exit the while loop and the algorithm returns the lists D and P . v1 v2 v3 v4 v5 (0, −) (∞, −) (∞, −) (∞, −) (∞, −) (10, v1 ) (3, v1 ) (11, v3 ) (5, v3 ) (7, v3 ) (9, v2 ) Table 3.4: Stepping through Dijkstra’s algorithm. 3.4. Dijkstra’s algorithm 125 Algorithm 3.5 A general template for Dijkstra’s algorithm. Input An undirected or directed graph G = (V, E) that is weighted and has no self- loops. The order of G is n > 0. A vertex s ∈ V from which to start the search. Vertices are numbered from 1 to n, i.e. V = {1, 2, . . . , n}. Output A list D of distances such that D[v] is the distance of a shortest path from s to v. A list P of vertex parents such that P [v] is the parent of v, i.e. v is adjacent from P [v]. 1: D ← [∞, ∞, . . . , ∞] . n copies of ∞ 2: D[s] ← 0 3: P ← [ ] 4: Q ← V . list of nodes to visit 5: while length(Q) > 0 do 6: find v ∈ Q such that D[v] is minimal 7: Q ← remove(Q, v) 8: for each u ∈ adj(v) ∩ Q do 9: if D[u] > D[v] + w(vu) then 10: D[u] ← D[v] + w(vu) 11: P [u] ← v 12: return (D, P ) 2 2 v2 v4 v2 v4 10 1 7 10 1 7 8 8 4 9 4 9 v1 v3 v5 v1 v3 v5 3 2 3 2 (a) Original digraph. (b) First iteration of while loop. 2 2 v2 v4 v2 v4 10 1 7 10 1 7 8 8 4 9 4 9 v1 v3 v5 v1 v3 v5 3 2 3 2 (c) Second iteration of while loop. (d) Third iteration of while loop. 2 2 v2 v4 v2 v4 10 1 7 8 4 9 4 v1 v3 v5 v1 v3 v5 3 2 3 2 (e) Fourth iteration of while loop. (f) Final shortest paths graph. Figure 3.10: Searching a weighted digraph using Dijkstra’s algorithm. 126 Chapter 3. Shortest paths algorithms Example 3.7. Apply Dijkstra’s algorithm to the graph in Figure 3.10(a), with starting vertex v1 . Solution. Dijkstra’s Algorithm 3.5 applied to the graph in Figure 3.10(a) yields the sequence of intermediary graphs shown in Figure 3.10, culminating in the final shortest paths graph of Figure 3.10(f) and Table 3.4. For any column vi in the table, each 2-tuple represents the distance and parent vertex of vi . As we move along the graph, processing vertices according to Dijkstra’s algorithm, the distance and parent vertex of a column are updated. The underlined 2-tuple represents the final distance and parent vertex produced by Dijkstra’s algorithm. From Table 3.4, we have the following shortest paths and distances: v1 -v2 : v1 , v3 , v2 d(v1 , v2 ) = 7 v1 -v3 : v1 , v3 d(v1 , v3 ) = 3 v1 -v4 : v1 , v3 , v2 , v4 d(v1 , v4 ) = 9 v1 -v5 : v1 , v3 , v5 d(v1 , v5 ) = 5 Intermediary vertices for a u-v path are obtained by starting from v and work backward using the parent of v, then the parent of the parent, and so on. Dijkstra’s algorithm is an example of a greedy algorithm. Whenever it tries to find the next vertex, it chooses only that vertex that minimizes the total weight so far. Greedy algorithms may not produce the best possible result. However, as the following theorem shows, Dijkstra’s algorithm does indeed produce shortest paths. Theorem 3.8. Correctness of Algorithm 3.5. Let G = (V, E) be a weighted (di)graph with a nonnegative weight function w. When Dijkstra’s algorithm is applied to G with source vertex s ∈ V , the algorithm terminates with D[u] = d(s, u) for all u ∈ V . Furthermore, if D[v] 6= ∞ and v 6= s, then s = u1 , u2 , . . . , uk = v is a shortest s-v path such that ui−1 = P [ui ] for i = 2, 3, . . . , k. Proof. If G is disconnected, then any v ∈ V that cannot be reached from s has distance D[v] = ∞ upon algorithm termination. Hence, it suffices to consider the case where G is connected. Let V = {s = v1 , v2 , . . . , vn } and use induction on i to show that after visiting vi we have D[v] = d(s, v) for all v ∈ Vi = {v1 , v2 , . . . , vi }. (3.6) For i = 1, equality holds. Assume for induction that (3.6) holds for some 1 ≤ i ≤ n − 1, so that now our task is to show that (3.6) holds for i + 1. To verify D[vi+1 ] = d(s, vi+1 ), note that by our inductive hypothesis, D[vi+1 ] = min {d(s, v) + w(vu) | v ∈ Vi and u ∈ adj(v) ∩ (Q\Vi )} and respectively D[vi+1 ] = min {d(s, v) + w(vu) | v ∈ Vi and u ∈ oadj(v) ∩ (Q\Vi )} if G is directed. Therefore, D[vi+1 ] = d(s, vi+1 ). Let v ∈ V such that D[v] 6= ∞ and v 6= s. We now construct an s-v path. When Algorithm 3.5 terminates, we have D[v] = D[v1 ] + w(v1 v), where P [v] = v1 and d(s, v) = d(s, v1 ) + w(v1 v). This means that v1 is the second-to-last vertex in a shortest s-v path. Repeated application of this process using the parent list P , we eventually produce a shortest s-v path s = vm , vm−1 , . . . , v1 , v, where P [vi ] = vi+1 for i = 1, 2, . . . , m − 1. 3.5. Bellman-Ford algorithm 127 To analyze the worst case time complexity of Algorithm 3.5, note that initializing D takes O(n + 1) and initializing Q takes O(n), for a total of O(n) devoted to initialization. Each extraction of a vertex v with minimal D[v] requires O(n) since we search through the entire list Q to determine the minimum value, for a total of O(n2 ). Each insertion into D requires constant time and the same holds for insertion into P . Thus, insertion into D and P takes O(|E| + |E|) = O(|E|), which require at most O(n) time. In the worst case, Dijkstra’s Algorithm 3.5 has running time O(n2 + n) = O(n2 ). Can we improve the run time of Dijkstra’s algorithm? The time complexity of Dijk- stra’s algorithm depends on its implementation. With a simple list implementation as presented in Algorithm 3.5, we have a worst case time complexity of O(n2 ), where n is the order of the graph under consideration. Let m be the size of the graph. Table 3.5 presents time complexities of Dijkstra’s algorithm for various implementations. Out of all the four implementations in this table, the heap implementations are much more efficient than the list implementation presented in Algorithm 3.5. A heap is a type of tree, a topic which will be covered in Chapter 2. Of all the heap implementations in Table 3.5, the Fibonacci heap implementation [82] yields the best runtime. Chapter 4 discusses how to use trees for efficient implementations of priority queues via heaps. Implementation Time complexity list O(n2 ) binary heap O (n + m) ln n k-ary heap O (kn + m) ln n ln k Fibonacci heap O(n ln n + m) Table 3.5: Implementation specific worst-case time complexity of Dijkstra’s algorithm. 3.5 Bellman-Ford algorithm — Randall Munroe, xkcd, http://xkcd.com/69/ A disadvantage of Dijkstra’s Algorithm 3.5 is that it cannot handle graphs with negative edge weights. The Bellman-Ford algorithm computes single-source shortest paths in a weighted graph or digraph, where some of the edge weights may be negative. This algorithm is a modification of the one published in 1957 by Richard E. Bellman [21] and that by Lester Randolph Ford, Jr. [79] in 1956. Shimbel [172] independently discovered the same method in 1955, and Moore [148] in 1959. In contrast to the “greedy” approach that Dijkstra’s algorithm takes, i.e. searching for the “cheapest” path, the Bellman-Ford algorithm searches over all edges and keeps track of the shortest one found as it searches. The Bellman-Ford Algorithm 3.6 runs in time O(mn), where m and n are the size and order of an input graph, respectively. To see this, note that the initialization on 128 Chapter 3. Shortest paths algorithms Algorithm 3.6 The Bellman-Ford algorithm. Input An undirected or directed graph G = (V, E) that is weighted and has no self- loops. Negative edge weights are allowed. The order of G is n > 0. A vertex s ∈ V from which to start the search. Vertices are numbered from 1 to n, i.e. V = {1, 2, . . . , n}. Output A list D of distances such that D[v] is the distance of a shortest path from s to v. A list P of vertex parents such that P [v] is the parent of v, i.e. v is adjacent from P [v]. If G has negative-weight cycles, then return false. Otherwise, return D and P . 1: D ← [∞, ∞, . . . , ∞] . n copies of ∞ 2: D[s] ← 0 3: P ← [ ] 4: for i ← 1, 2, . . . , n − 1 do 5: for each edge uv ∈ E do 6: if D[v] > D[u] + w(uv) then 7: D[v] ← D[u] + w(uv) 8: P [v] ← u 9: for each edge uv ∈ E do 10: if D[v] > D[u] + w(uv) then 11: return false 12: return (D, P ) lines 1 to 3 takes O(n). Each of the n − 1 rounds of the for loop starting on line 4 takes O(m), for a total of O(mn) time. Finally, the for loop starting on line 9 takes O(m). The loop starting on line 4 performs at most n − 1 updates of the distance D[v] of each head of an edge. Many graphs have sizes that are less then n − 1, resulting in a number of redundant rounds of updates. To avoid such redundancy, we could add an extra check in the outer loop spanning lines 4 to 8 to immediately terminate that outer loop after any round that did not result in an update of any D[v]. Algorithm 3.7 presents a modification of the Bellman-Ford Algorithm 3.6 that avoids redundant rounds of updates. 3.6 Floyd-Roy-Warshall algorithm The shortest distance between two points is not a very interesting journey. — R. Goldberg Let D be a weighted digraph of order n and size m. Dijkstra’s Algorithm 3.5 and the Bellman-Ford Algorithm 3.6 can be used to determine shortest paths from a single source vertex to all other vertices of D. To determine a shortest path between each pair of distinct vertices in D, we repeatedly apply either of these algorithms to each vertex of D. Such repeated application of Dijkstra’s and the Bellman-Ford algorithms results in algorithms that run in time O(n3 ) and O(n2 m), respectively. The Floyd-Roy-Warshall algorithm (FRW), or the Floyd-Warshall algorithm, is an algorithm for finding shortest paths in a weighted, directed graph. Like the Bellman- Ford algorithm, it allows for negative edge weights and detects a negative weight cycle if one exists. Assuming that there are no negative weight cycles, a single execution of 3.6. Floyd-Roy-Warshall algorithm 129 Algorithm 3.7 The Bellman-Ford algorithm with checks for redundant updates. Input An undirected or directed graph G = (V, E) that is weighted and has no self- loops. Negative edge weights are allowed. The order of G is n > 0. A vertex s ∈ V from which to start the search. Vertices are numbered from 1 to n, i.e. V = {1, 2, . . . , n}. Output A list D of distances such that D[v] is the distance of a shortest path from s to v. A list P of vertex parents such that P [v] is the parent of v, i.e. v is adjacent from P [v]. If G has negative-weight cycles, then return false. Otherwise, return D and P . 1: D ← [∞, ∞, . . . , ∞] . n copies of ∞ 2: D[s] ← 0 3: P ← [ ] 4: for i ← 1, 2, . . . , n − 1 do 5: updated ← false 6: for each edge uv ∈ E do 7: if D[v] > D[u] + w(uv) then 8: D[v] ← D[u] + w(uv) 9: P [v] ← u 10: updated ← true 11: if updated = false then 12: exit the loop 13: for each edge uv ∈ E do 14: if D[v] > D[u] + w(uv) then 15: return false 16: return (D, P ) 130 Chapter 3. Shortest paths algorithms the FRW algorithm will find the shortest paths between all pairs of vertices. It was discovered independently by Bernard Roy [167] in 1959, Robert Floyd [78] in 1962, and by Stephen Warshall [189] in 1962. In some sense, the FRW algorithm is an example of dynamic programming, which allows one to break the computation into simpler steps using some sort of recursive procedure. The rough idea is as follows. Temporarily label the vertices of a weighted digraph G as V = {1, 2, . . . , n} with n = |V (G)|. Let W = [w(i, j)] be the weight matrix of G where w(ij), if ij ∈ E(G), w(i, j) = 0, if i = j, (3.7) ∞, otherwise. Let Pk (i, j) be a shortest path from i to j such that its intermediate vertices are in {1, 2, . . . , k}. Let Dk (i, j) be the weight (or distance) of Pk (i, j). If no shortest i-j paths exist, define Pk (i, j) = ∞ and Dk (i, j) = ∞ for all k ∈ {1, 2, . . . , n}. If k = 0, then P0 (i, j) : i, j since no intermediate vertices are allowed in the path and hence D0 (i, j) = w(i, j). In other words, if i and j are adjacent, a shortest i-j path is the edge ij itself and the weight of this path is simply the weight of ij. Now consider Pk (i, j) for k > 0. Either Pk (i, j) passes through k or it does not. If k is not on the path Pk (i, j), then the intermediate vertices of Pk (i, j) are in {1, 2, . . . , k − 1}, as are the vertices of Pk−1 (i, j). In case Pk (i, j) contains the vertex k, then Pk (i, j) traverses k exactly once by the definition of path. The i-k subpath in Pk (i, j) is a shortest i-k path whose intermediate vertices are drawn from {1, 2, . . . , k − 1}, which is also the set of intermediate vertices for the k-j subpath in Pk (i, j). That is, to obtain Pk (i, j), we take the union of the paths Pk−1 (i, k) and Pk−1 (k, j). We compute the weight Dk (i, j) of Pk (i, j) using the expression ( w(i, j), if k = 0, Dk (i, j) = (3.8) min{Dk−1 (i, j), Dk−1 (i, k) + Dk−1 (k, j)}, if k > 0. The key to the Floyd-Roy-Warshall algorithm lies in exploiting expression (3.8). If n = |V |, then this is a O(n3 ) time algorithm. For comparison, the Bellman-Ford al- gorithm has complexity O(|V | · |E|), which is O(n3 ) time for dense graphs. However, Bellman-Ford only yields the shortest paths emanating from a single vertex. To achieve comparable output, we would need to iterate Bellman-Ford over all vertices, which would be an O(n4 ) time algorithm for dense graphs. Except possibly for sparse graphs, Floyd- Roy-Warshall is better than an iterated implementation of Bellman-Ford. Note that Pk (i, k) = Pk−1 (i, k) and Pk (k, i) = Pk−1 (k, i), consequently Dk (i, k) = Dk−1 (i, k) and Dk (k, i) = Dk−1 (k, i). This observation allows us to replace Pk (i, j) with P (i, j) for k = 1, 2, . . . , n. The final results of P (i, j) and D(i, k) are the same as Pn (i, j) and Dn (i, j), respectively. Algorithm 3.8 summarizes the above discussion into an algorith- mic presentation. Like the Bellman-Ford algorithm, the Floyd-Roy-Warshall algorithm can also detect the presence of negative weight cycles. If G is a weighted digraph without self-loops, by (3.7) we have D(i, i) = 0 for i = 1, 2, . . . , n. Any path p starting and ending at i could only improve upon the initial weight of 0 if the weight sum of p is less than zero, i.e. a negative weight cycle. Upon termination of Algorithm 3.8, if D(i, i) < 0, we conclude that there is a path starting and ending at i whose weight sum is negative. 3.6. Floyd-Roy-Warshall algorithm 131 Algorithm 3.8 The Floyd-Roy-Warshall algorithm for all-pairs shortest paths. Input A weighted digraph G = (V, E) that has no self-loops. Negative edge weights are allowed. The order of G is n > 0. Vertices are numbered from 1 to n, i.e. V = {1, 2, . . . , n}. The weight matrix W = [w(i, j)] of G as defined in (3.7). Output A matrix P = [aij ] of shortest paths in G. A matrix D = [aij ] of distances where D[i, j] is the weight (or distance) of a shortest i-j path in G. 1: n ← |V | 2: P [aij ] ← an n × n zero matrix 3: D[aij ] ← W [w(i, j)] 4: for k ← 1, 2, . . . , n do 5: for i ← 1, 2, . . . , n do 6: for j ← 1, 2, . . . , n do 7: if D[i, j] > D[i, k] + D[k, j] then 8: P [i, j] ← k 9: D[i, j] ← D[i, k] + D[k, j] 10: return (P, D) Here is an implementation in Sage. def fl oy d_roy_warshall ( A ): """ Shortest paths INPUT : - A -- weighted adjacency matrix OUTPUT : - dist -- a matrix of distances of shortest paths . - paths -- a matrix of shortest paths . """ G = Graph (A , format = " w e i g h t e d _ a d j a c e n c y _ m a tr i x " ) V = G . vertices () E = [( e [0] , e [1]) for e in G . edges ()] n = len ( V ) dist = [[0]* n for i in range ( n )] paths = [[ -1]* n for i in range ( n )] # initialization step for i in range ( n ): for j in range ( n ): if (i , j ) in E : paths [ i ][ j ] = j if i == j : dist [ i ][ j ] = 0 elif A [ i ][ j ] < >0: dist [ i ][ j ] = A [ i ][ j ] else : dist [ i ][ j ] = infinity # iteratively finding the shortest path for j in range ( n ): for i in range ( n ): if i <> j : for k in range ( n ): if k <> j : if dist [ i ][ k ] > dist [ i ][ j ]+ dist [ j ][ k ]: paths [ i ][ k ] = V [ j ] dist [ i ][ k ] = min ( dist [ i ][ k ] , dist [ i ][ j ] + dist [ j ][ k ]) for i in range ( n ): if dist [ i ][ i ] < 0: raise ValueError , " A negative edge weight cycle exists . " return dist , matrix ( paths ) Here are some examples. sage : A = matrix ([[0 ,1 ,2 ,3] , [0 ,0 ,2 ,1] , [ -5 ,0 ,0 ,3] , [1 ,0 ,1 ,0]]); A 132 Chapter 3. Shortest paths algorithms sage : f lo yd_roy_warshall ( A ) Traceback ( click to the left of this block for traceback ) ... ValueError : A negative edge weight cycle exists . The plot of this weighted digraph with four vertices appears in Figure 3.11. 1 3 3 2 2 3 1 2 1 5 0 1 1 Figure 3.11: Demonstrating the Floyd-Roy-Warshall algorithm. sage : A = matrix ([[0 ,1 ,2 ,3] , [0 ,0 ,2 ,1] , [ -1/2 ,0 ,0 ,3] , [1 ,0 ,1 ,0]]); A sage : f lo yd_roy_warshall ( A ) ([[0 , 1 , 2 , 2] , [3/2 , 0 , 2 , 1] , [ -1/2 , 1/2 , 0 , 3/2] , [1/2 , 3/2 , 1 , 0]] , [ -1 1 2 1] [ 2 -1 2 3] [ -1 0 -1 1] [ 2 2 -1 -1]) The plot of this weighted digraph with four vertices appears in Figure 3.12. 1 1 2 3 3 1 1 2 1 −0.5 3 2 0 Figure 3.12: Another demonstration of the Floyd-Roy-Warshall algorithm. Example 3.9. Section ?? briefly presented the concept of molecular graphs in chem- istry. The Wiener number of a molecular graph was first published in 1947 by Harold 3.6. Floyd-Roy-Warshall algorithm 133 Figure 3.13: Molecular graph of 1,1,3-trimethylcyclobutane. Wiener [196] who used it in chemistry to study properties of alkanes. Other applica- tions [94] of the Wiener number to chemistry are now known. If G = (V, E) is a connected graph with vertex set V = {v1 , v2 , . . . , vn }, then the Wiener number W of G is defined by X W (G) = d(vi , vj ) (3.9) i<j where d(vi , vj ) is the distance from vi to vj . What is the Wiener number of the molecular graph in Figure 3.13? Solution. Consider the molecular graph in Figure 3.13 as directed with unit weight. To compute the Wiener number of this graph, use the Floyd-Roy-Warshall algorithm to obtain a distance matrix D = [di,j ], where di,j is the distance from vi to vj , and apply the definition (3.9). The distance matrix resulting from the Floyd-Roy-Warshall algorithm is 0 2 1 2 3 2 4 2 0 1 2 3 2 4 1 1 0 1 2 1 3 M = 2 2 1 0 1 2 2 . 3 3 2 1 0 1 1 2 2 1 2 1 0 2 4 4 3 2 1 2 0 Sum all entries in the upper (or lower) triangular of M to obtain the Wiener number W = 42. Using Sage, we have sage : G = Graph ({1:[3] , 2:[3] , 3:[4 ,6] , 4:[5] , 6:[5] , 5:[7]}) sage : D = G . sh or t es t _p at h _a l l_ pa i rs ()[0] sage : M = [ D [ i ][ j ] for i in D for j in D [ i ]] sage : M = matrix (M , nrows =7 , ncols =7) sage : W = 0 sage : for i in range ( M . nrows () - 1): ... for j in range ( i +1 , M . ncols ()): ... W += M [i , j ] sage : W 42 which verifies our computation above. See [94] for a survey of some results concerning the Wiener number. 3.6.1 Transitive closure Consider a digraph G = (V, E) of order n = |V |. The transitive closure of G is defined as the digraph G∗ = (V, E ∗ ) having the same vertex set as G. However, the edge set 134 Chapter 3. Shortest paths algorithms E ∗ of G∗ consists of all edges uv such that there is a u-v path in G and uv ∈ / E. The ∗ transitive closure G answers an important question about G: If u and v are two distinct vertices of G, are they connected by a path with length ≥ 1? To compute the transitive closure of G, we let each edge of G be of unit weight and apply the Floyd-Roy-Warshall Algorithm 3.8 on G. By Proposition 1.13, for any i-j path in G we have D[i, j] < n, and if there are no paths from i to j in G, we have D[i, j] = ∞. This procedure for computing transitive closure runs in time O(n3 ). Modifying the Floyd-Roy-Warshall algorithm slightly, we obtain an algorithm for computing transitive closure that, in practice, is more efficient than Algorithm 3.8 in terms of time and space. Instead of using the operations min and + as is the case in the Floyd-Roy-Warshall algorithm, we replace these operations with the logical operations ∨ (logical OR) and ∧ (logical AND), respectively. For i, j, k = 1, 2, . . . , n, define Tk (i, j) = 1 if there is an i-j path in G with all intermediate vertices belonging to {1, 2, . . . , k}, and Tk (i, j) = 0 otherwise. Thus, the edge ij belongs to the transitive closure G∗ if and only if Tk (i, j) = 1. The definition of Tk (i, j) can be cast in the form of a recursive definition as follows. For k = 0, we have ( 0, if i 6= j and ij ∈ / E, T0 (i, j) = 1, if i = j or ij ∈ E and for k > 0, we have Tk (i, j) = Tk−1 (i, j) ∨ Tk−1 (i, k) ∧ Tk−1 (k, j) . We need not use the subscript k at all and instead let T be a boolean matrix such that T [i, j] = 1 if and only if there is an i-j path in G, and T [i, j] = 0 otherwise. Using the above notations, the Floyd-Roy-Warshall algorithm is translated to Algorithm 3.9 for obtaining the boolean matrix T . We can then use T and the definition of transitive closure to obtain the edge set E ∗ in the transitive closure G∗ = (V, E ∗ ) of G = (V, E). A more efficient transitive closure algorithm can be found in the PhD thesis of Esko Nuutila [157]. See also the method of four Russians [1,8]. The transitive closure algorithm as presented in Algorithm 3.9 is due to Stephen Warshall [189]. It is a special case of a more general algorithm in automata theory due to Stephen Kleene [119], called Kleene’s algorithm. Algorithm 3.9 Variant of the Floyd-Roy-Warshall algorithm for transitive closure. Input A digraph G = (V, E) that has no self-loops. Vertices are numbered from 1 to n, i.e. V = {1, 2, . . . , n}. Output The boolean matrix T such that T [i, j] = 1 if and only if there is an i-j path in G, and T [i, j] = 0 otherwise. 1: n ← |V | 2: T ← adjacency matrix of G 3: for k ← 1, 2, . . . , n do 4: for i ← 1, 2, . . . , n do 5: for j ← 1, 2, . . . , n do 6: T [i, j] ← T [i, j] ∨ T [i, k] ∧ T [k, j] 7: return T 3.7. Johnson’s algorithm 135 3.7 Johnson’s algorithm The shortest distance between two points is under construction. — Noelie Altito Let G = (V, E) be a sparse digraph with edge weights but no negative cycles. Johnson’s algorithm [111] finds a shortest path between each pair of vertices in G. First published in 1977 by Donald B. Johnson, the main insight of Johnson’s algorithm is to combine the technique of edge reweighting with the Bellman-Ford and Dijkstra’s algorithms. The Bellman-Ford algorithm is first used to ensure that G has no negative cycles. Next, we reweight edges in such a manner as to preserve shortest paths. The final stage makes use of Dijkstra’s algorithm for computing shortest paths between all vertex pairs. Pseudocode for Johnson’s algorithm is presented in Algorithm 3.10. With a Fibonacci heap implementation of the minimum-priority queue, the time complexity for sparse graphs is O(|V |2 log |V | + |V | · |E|), where n = |V | is the number of vertices of the original graph G. To prove the correctness of Algorithm 3.10, we need to show that the new set of edge weights produced by ŵ must satisfy two properties: 1. The reweighted edges preserve shortest paths. That is, let p be a u-v path for u, v ∈ V . Then p is a shortest weighted path using weight function w if and only if p is also a shortest weighted path using weight function ŵ. 2. The reweight function ŵ produces nonnegative weights. In other words, if u, v ∈ V then ŵ(uv) ≥ 0. Both of these properties are proved in Lemma 3.10. Algorithm 3.10 Johnson’s algorithm for sparse graphs. Input A sparse weighted digraph G = (V, E), where the vertex set is V = {1, 2, . . . , n}. Output If G has negative-weight cycles, then return false. Otherwise, return an n × n matrix D of shortest-path weights and a list P such that P [v] is a parent list resulting from running Dijkstra’s algorithm on G with start vertex v. 1: s ← vertex not in V 2: V 0 ← V ∪ {s} 3: E 0 ← E ∪ {sv|v ∈ V } 4: G0 ← digraph (V 0 , E 0 ) with weight w(sv) = 0 for all v ∈ V 5: if BellmanFord(G0 , w, s) = false then 6: return false 7: d ← distance list returned by BellmanFord(G0 , w, s) 8: for each edge uv ∈ E 0 do 9: ŵ(uv) ← w(uv) + d[u] − d[v] 10: for each u ∈ V do 11: (δ̂, P̂ ) ← distance and parent lists returned by Dijkstra(G, ŵ, u) 12: P [u] ← P̂ 13: for each v ∈ V do 14: D[u, v] ← δ̂[v] + d[v] − d[u] 15: return (D, P ) 136 Chapter 3. Shortest paths algorithms Lemma 3.10. Reweighting preserves shortest paths. Let G = (V, E) be a weighted digraph having weight function w : E → R and let h : V → R be a mapping of vertices to real numbers. Let ŵ be another weight function for G such that ŵ(uv) = w(uv) + h(u) − h(v) for all uv ∈ E. Suppose p : v0 , v1 , . . . , vk is any path in G. Then we have the following results. 1. The path p is a shortest v0 -vk path with weight function w if and only if it is a shortest v0 -vk path with weight function ŵ. 2. The graph G has a negative cycle using weight function w if and only if G has a negative cycle using ŵ. 3. If G has no negative cycles, then ŵ(uv) ≥ 0 for all uv ∈ E. Proof. Write δ and δ̂ for the shortest path weights derived from w and ŵ, respectively. To prove part 1, we need to show that w(p) = δ(v0 , vk ) if and only if ŵ(p) = δ̂(v0 , vk ). First, note that any v0 -vk path p satisfies ŵ(p) = w(p) + h(v0 ) − h(vk ) because k X ŵ(p) = ŵ(vi−1 vi ) i=1 Xk = w(vi−1 vi ) + h(vi−1 ) − h(vi ) i=1 Xk k X = w(vi−1 vi ) + h(vi−1 ) − h(vi ) i=1 i=1 Xk = w(vi−1 vi ) + h(v0 ) − h(vk ) i=1 = w(p) + h(v0 ) − h(vk ). Any v0 -vk path shorter than p and using weight function w is also shorter using ŵ. Therefore, w(p) = δ(v0 , vk ) if and only if ŵ(p) = δ̂(v0 , vk ). To prove part 2, consider any cycle c : v0 , v1 , . . . , vk where v0 = vk . Using the proof of part 1, we have ŵ(c) = w(c) + h(v0 ) − h(vk ) = w(c) thus showing that c is a negative cycle using w if and only if it is a negative cycle using ŵ. To prove part 3, we construct a new graph G0 = (V 0 , E 0 ) as follows. Consider a vertex s∈ / V and let V 0 = V ∪ {s} and E 0 = E ∪ {sv | v ∈ V }. Extend the weight function w to include w(sv) = 0 for all v ∈ V . By construction, s has no incoming edges and any path in G0 that contains s has s as the source vertex. Thus G0 has no negative cycles if and only if G has no negative cycles. Define the function h : V → R by v 7→ δ(s, v) with domain V 0 . By the triangle inequality (see Lemma 3.5), δ(s, u) + w(uv) ≥ δ(s, v) ⇐⇒ h(u) + w(uv) ≥ h(v) thereby showing that ŵ(uv) = w(uv) + h(u) − h(v) ≥ 0. 3.8. Problems 137 3.8 Problems I believe that a scientist looking at nonscientific problems is just as dumb as the next guy. — Richard Feynman 3.1. Let G = (V, E) be an undirected graph, let s ∈ V , and D is the list of distances resulting from running Algorithm 3.1 with G and s as input. Show that G is connected if and only if D[v] is defined for each v ∈ V . 3.2. Show that the worst-case time complexity of depth-first search Algorithm 3.2 is O(|V | + |E|). 3.3. Let D be the list of distances returned by Algorithm 3.2, let s be a starting vertex, and let v be a vertex such that D[v] 6= ∞. Show that D[v] is the length of any shortest path from s to v. 3.4. Consider the graph in Figure 3.10 as undirected. Run this undirected version through Dijkstra’s algorithm with starting vertex v1 . 1 v3 v4 2 3 1 1 1 2 3 v5 3 6 v2 v1 1 Figure 3.14: Searching a directed house graph using Dijkstra’s algorithm. 3.5. Consider the graph in Figure 3.14. Choose any vertex as a starting vertex and run Dijkstra’s algorithm over it. Now consider the undirected version of that digraph and repeat the exercise. 3.6. For each vertex v of the graph in Figure 3.14, run breadth-first search over that graph with v as the starting vertex. Repeat the exercise for depth-first search. Compare the graphs resulting from the above exercises. 3.7. A list data structure can be used to manage vertices and edges. If L is a nonempty list of vertices of a graph, we may want to know whether the graph contains a particular vertex. We could search the list L, returning True if L contains the vertex in question and False otherwise. Linear search is a simple searching algorithm. Given an object E for which we want to search, we consider each element e of L in turn and compare E to e. If at any point during our search we found a match, we halt the algorithm and output an affirmative answer. Otherwise, we have scanned through all elements of L and each of those elements do not match E. In this case, linear search reports that E is not in L. Our discussion is summarized in Algorithm 3.11. 138 Chapter 3. Shortest paths algorithms (a) Implement Algorithm 3.11 in Sage and test your implementation using the graphs presented in the figures of this chapter. (b) What is the maximum number of comparisons during the running of Algo- rithm 3.11? What is the average number of comparisons? (c) Why must the input list L be nonempty? Algorithm 3.11 Linear search for lists. Input A nonempty list L of vertices or edges. An object E for which we want to search in L. Output true if E is in L; false otherwise. 1: for each e ∈ L do 2: if E = e then 3: return true 4: return false Algorithm 3.12 Binary search for lists of positive integers. Input A nonempty list L of positive integers. Elements of L are sorted in nondecreasing order. An integer i for which we want to search in L. Output true if i is in L; false otherwise. 1: low ← 0 2: high ← |L| − 1 3: while low ≤ high do 4: mid ← b low +2 high c 5: if i = L[mid] then 6: return true 7: if i < L[mid] then 8: high ← mid − 1 9: else 10: low ← mid + 1 11: return false 3.8. Binary search is a much faster searching algorithm than linear search. The binary search algorithm assumes that its input list is ordered in some manner. For sim- plicity, we assume that the input list L consists of positive integers. The main idea of binary search is to partition L into two halves: the left half and the right half. Our task now is to determine whether the object E of interest is in the left half or the right half, and apply binary search recursively to the half in which E is located. Algorithm 3.12 provides pseudocode of our discussion of binary search. (a) Implement Algorithm 3.12 in Sage and test your implementation using the graphs presented in the figures of this chapter. (b) What is the worst case runtime of Algorithm 3.12? How does this compare to the worst case runtime of linear search? (c) Why must the input list L be sorted in nondecreasing order? Would Algo- rithm 3.12 work if L is sorted in nonincreasing order? If not, modify Algo- rithm 3.12 so that it works with an input list that is sorted in nonincreasing order. 3.8. Problems 139 (d) Line 4 of Algorithm 3.12 uses the floor function to compute the index of the middle value. Would binary search still work if we use the ceiling function instead of the floor function? 3.9. Let G be a simple undirected graph having distance matrix D = [d(vi , vj )], where d(vi , vj ) ∈ R denotes the shortest distance from vi ∈ V (G) to vj ∈ V (G). If vi = vj , we set d(vi , vj ) = 0. For each pair of distinct vertices (vi , vj ), we have d(vi , vj ) = d(vj , vi ). The i-j entry of D is also written as di,j and denotes the entry in row i and column j. (a) The total distance td(u) of a fixed vertex u ∈ V (G) is the sum of distances from u to each vertex in G: X td(u) = d(u, v). v∈V (G) If G is connected, i is the row index of vertex u in the distance matrix D, and j is the column index of u in D, show that the total distance of u is X X td(u) = di,k = dk,j . (3.10) k k (b) Let the vertices of G be labeled V = {v1 , v2 , . . . , vn }, where n = |V (G)| is the order of G. The total distance td(G) of G is obtained by summing all the d(vi , vj ) for i < j. If G is connected, show that the total distance of G is equal to the sum of all entries in the upper (or lower) triangular of D: ! X X 1 XX td(G) = di,j = di,j = d(u, v) . (3.11) i<j i>j 2 u∈V v∈V Hence show that the total distance of G is equal to its Wiener number: td(G) = W (G). (c) Would equations (3.10) and (3.11) hold if G is not connected or directed? 3.10. The following result is from [202]. Let G1 and G2 be graphs with orders ni = |V (Gi )| and sizes mi = |E(Gi )|, respectively. (a) If each of G1 and G2 is connected, show that the Wiener number of the Cartesian product G1 G2 is W (G1 G2 ) = n22 · W (G1 ) + n21 · W (G2 ). (b) If G1 and G2 are arbitrary graphs, show that the Wiener number of the join G1 + G2 is W (G1 + G2 ) = n21 − n1 + n22 − n2 + n1 n2 − m1 − m2 . 3.11. The following results originally appeared in [70] and independently rediscovered many times since. 140 Chapter 3. Shortest paths algorithms (a) If Pn is the path graph on n ≥ 0 vertices, show that the Wiener number of Pn is W (Pn ) = 16 n(n2 − 1). (b) If Cn is the cycle graph on n ≥ 0 vertices, show that the Wiener number of Cn is (1 8 n(n2 − 1), if n is odd, W (Cn ) = 1 3 8 n, if n is even. (c) If Kn is the complete graph on n vertices, show that its Wiener number is W (Kn ) = 21 n(n − 1). (d) Show that the Wiener number of the complete bipartite graph Km,n is W (Km,n ) = mn + m(m − 1) + n(n − 1). 3.12. In addition to searching, there is the related problem of sorting a list according to an ordering relation. If the given list L = [e1 , e2 , . . . , en ] consists of real numbers, we want to order elements in nondecreasing order. Bubble sort is a basic sorting algorithm that can be used to sort a list of real numbers, indeed any collection of objects that can be ordered according to an ordering relation. During each pass through the list L from left to right, we consider ei and its right neighbor ei+1 . If ei ≤ ei+1 , then we move on to consider ei+1 and its right neighbor ei+2 . If ei > ei+1 , then we swap these two values around in the list and then move on to consider ei+1 and its right neighbor ei+2 . Each successive pass pushes to the right end an element that is the next largest in comparison to the previous largest element pushed to the right end. Hence the name bubble sort for the algorithm. Algorithm 3.13 summarizes our discussion. Algorithm 3.13 Bubble sort. Input A list L of n > 1 elements that can be ordered using the “less than or equal to” relation “≤”. Output The same list as L, but sorted in nondecreasing order. 1: for i ← n, n − 1, . . . , 2 do 2: for j ← 2, 3, . . . , i do 3: if L[j − 1] > L[j] then 4: swap the values of L[j − 1] and L[j] 5: return L (a) Analyze the worst-case runtime of Algorithm 3.13. (b) Modify Algorithm 3.13 so that it sorts elements in nonincreasing order. 3.13. Selection sort is another simple sorting algorithm that works as follows. Let L = [e1 , e2 , . . . , en ] be a list of elements that can be ordered according to the relation “≤”, e.g. the ei can all be real numbers or integers. On the first scan of L from left to right, among the elements L[2], . . . , L[n] we find the smallest element and exchange it with L[1]. On the second scan, we find the smallest element among L[3], . . . , L[n] and exchange that smallest element with L[2]. In general, during the i-th scan we find the smallest element among L[i + 1], . . . , L[n] and exchange that with L[i]. At the end of the i-th scan, the element L[i] is in its final 3.8. Problems 141 position and would not be processed again. When the index reaches i = n, the list would have been sorted in nondecreasing order. The procedure is summarized in Algorithm 3.14. (a) Analyze the worst-case runtime of Algorithm 3.14 and compare your result to the worst-case runtime of the bubble sort Algorithm 3.13. (b) Modify Algorithm 3.14 to sort elements in nonincreasing order. (c) Line 6 of Algorithm 3.14 assumes that among L[i+1], L[i+2], . . . , L[n] there is a smallest element L[k] such that L[i] > L[k], hence we perform the swap. It is possible that L[i] < L[k], obviating the need to carry out the value swapping. Modify Algorithm 3.14 to take account of our discussion. Algorithm 3.14 Selection sort. Input A list L of n > 1 elements that can be ordered using the relation “≤”. Output The same list as L, but sorted in nondecreasing order. 1: for i ← 1, 2, . . . , n − 1 do 2: min ← i 3: for j ← i + 1, i + 2, . . . , n do 4: if L[j] < L[min] then 5: min ← j 6: swap the values of L[min] and L[i] 7: return L 3.14. Algorithm 3.3 uses breadth-first search to determine the connectivity of an undi- rected graph. Modify this algorithm to use depth-first search. How can Algo- rithm 3.3 be used or modified to test the connectivity of a digraph? 3.15. The following problem is known as the river crossing problem. A man, a goat, a wolf, and a basket of cabbage are all on one side of a river. They have a boat that could be used to cross to the other side of the river. The boat can only hold at most two passengers, one of whom must be able to row the boat. One of the two passengers must be the man and the other passenger can be either the goat, the wolf, or the basket of cabbage. When crossing the river, if the man leaves the wolf with the goat, the wolf would prey on the goat. If he leaves the goat with the basket of cabbage, the goat would eat the cabbage. The objective is to cross the river in such a way that the wolf has no chance of preying on the goat, nor that the goat eat the cabbage. (a) Let M , G, W , and C denote the man, the goat, the wolf, and the basket of cabbage, respectively. Initially all four are on the left side of the river and none of them are on the right side. Denote this by the ordered pair (M GW C, w), which is called the initial state of the problem. When they have all crossed to the right side of the river, the final state of the problem is (w, M GW C). The underscore “w” means that neither M , G, W , nor C are on the corresponding side of the river. List a finite sequence of moves to get from (M GW C, w) to (w, M GW C). Draw your result as a digraph. (b) In the digraph Γ obtained from the previous exercise, let each edge of Γ be of unit weight. Find a shortest path from (M GW C, w) to (w, M GW C). 142 Chapter 3. Shortest paths algorithms (c) Rowing from one side of the river to the other side is called a crossing. What is the minimum number of crossings needed to get from (M GW C, w) to (w, M GW C)? 3.16. Symbolic computation systems such as Magma, Maple, Mathematica, Maxima, and Sage are able to read in a symbolic expression such as (a + b)^2 - (a - b)^2 and determine whether or not the brackets match. A bracket is any of the following characters: ( ) { } [ ] A string S of characters is said to be balanced if any left bracket in S has a corre- sponding right bracket that is also in S. Furthermore, if there are k occurrences of one type of left bracket, then there must be k occurrences of the corresponding right bracket. The balanced bracket problem is concerned with determining whether or not the brackets in S are balanced. Algorithm 3.15 contains a procedure to de- termine if the brackets in S are balanced, and if so return a list of positive integers to indicate how the brackets match. (a) Implement Algorithm 3.15 in Sage and test your implementation on various strings containing brackets. Test your implementation on nonempty strings without any brackets. (b) Modify Algorithm 3.15 so that it returns True if the brackets of an input string are balanced, and returns False otherwise. (c) What is the worst-case runtime of Algorithm 3.15? 3.17. An arithmetic expression written in the form a + b is said to be in infix notation because the operator is in between the operands. The same expression can also be written in reverse Polish notation (or postfix notation) as ab+ with the operator following its two operands. Given an arithmetic expression A = e0 e1 · · · en written in reverse Polish notation, we can use the stack data structure to evaluate the expression. Let P = [e0 , e1 , . . . , en ] be the stack representation of A, where traversing P from left to right we are moving from the top of the stack to the bottom of the stack. We call P the Polish stack and the stack E containing intermediate results the evaluation stack. While P is not empty, pop the Polish stack and assign the extracted result to x. If x is an operator, we pop the evaluation stack twice: the result of the first pop is assigned to b and the result of the second pop is assigned to a. Compute the infix expression a x b and push the result onto E. However, if x is an operand, we push x onto E. Iterate the above process until P is empty, at which point the top of E contains the evaluation of A. Refer to Algorithm 3.16 for pseudocode of the above discussion. (a) Prove the correctness of Algorithm 3.16. (b) What is the worst-case runtime of Algorithm 3.16? 3.8. Problems 143 Algorithm 3.15 A brackets parser. Input A nonempty string S of characters. Output A list L of positive integers indicating how the brackets match. If the brackets are not balanced, return the empty string ε. 1: L ← [ ] 2: T ← empty stack 3: c ← 1 4: n ← |S| 5: for i ← 0, 1, . . . , n do 6: if S[i + 1] is a left bracket then 7: append(L, c) 8: push (S[i + 1], c) onto T 9: c←c+1 10: if S[i + 1] is a right bracket then 11: if T is empty then 12: return ε 13: (left, d) ← pop(T ) 14: if left matches S[i + 1] then 15: append(L, d) 16: else 17: return ε 18: if T is empty then 19: return L 20: return ε 144 Chapter 3. Shortest paths algorithms Algorithm 3.16 Evaluate arithmetic expressions in reverse Polish notation. Input A Polish stack P containing an arithmetic expression in reverse Polish notation. Output An evaluation of the arithmetic expression represented by P . 1: E ← empty stack 2: v ← Null 3: while P is not empty do 4: x ← pop(P ) 5: if x is an operator then 6: b ← pop(E) 7: a ← pop(E) 8: if x is addition operator then 9: v ←a+b 10: else if x is subtraction operator then 11: v ←a−b 12: else if x is multiplication operator then 13: v ←a×b 14: else if x is division operator then 15: v ← a/b 16: else 17: exit algorithm with error 18: push(E, v) 19: else 20: push(E, x) 21: v ← pop(E) 22: return v 3.8. Problems 145 (c) Modify Algorithm 3.16 to support the exponentiation operator. 3.18. Figure 3.5 provides a knight’s tour for the knight piece with initial position as in Figure 3.5(a). By rotating the chessboard in Figure 3.5(b) by 90n degrees for positive integer values of n, we obtain another knight’s tour that, when represented as a graph, is isomorphic to the graph in Figure 3.5(c). (a) At the beginning of the 18th century, de Montmort and de Moivre provided the following strategy [12, p.176] to solve the knight’s tour problem on an 8 × 8 chessboard. Divide the board into an inner 4 × 4 square and an outer shell of two squares deep, as shown in Figure 3.15(a). Place a knight on a square in the outer shell and move the knight piece around that shell, always in the same direction, so as to visit each square in the outer shell. After that, move into the inner square and solve the knight’s tour problem for the 4 × 4 case. Apply this strategy to solve the knight’s tour problem with the initial position as in Figure 3.15(b). (b) Use the Montmort-Moivre strategy to obtain a knight’s tour, starting at the position of the black-filled node in the outer shell in Figure 3.5(b). (c) A re-entrant or closed knight’s tour is a knight’s tour that starts and ends at the same square. Find re-entrant knight’s tours with initial positions as in Figure 3.16. (d) Devise a backtracking algorithm to solve the knight’s tour problem on an n×n chessboard for n > 3. 0Z0Z0Z0Z 0Z0Z0Z0m Z0Z0Z0Z0 Z0Z0Z0Z0 0Z0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0Z0 Z0Z0Z0Z0 0Z0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0Z0 Z0Z0Z0Z0 0Z0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0Z0 Z0Z0Z0Z0 (a) A 4 × 4 inner square. (b) Initial position in the outer shell. Figure 3.15: De Montmort and de Moivre’s solution strategy for the 8 × 8 knight’s tour problem. 3.19. The n-queens problem is concerned with the placement of n queens on an n × n chessboard such that no two queens can attack each other. Two queens can attack each other if they are in the same row, column, diagonal, or antidiagonal of the chessboard. The trivial case n = 1 is easily solved by placing the one queen in the only given position. There are no solutions for the cases n = 2, 3. Solutions for the cases n = 4, 8 are shown in Figure 3.17. Devise a backtracking algorithm to 146 Chapter 3. Shortest paths algorithms 0Z0Z0Z0Z Z0Z0Z0Z0 0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0 Z0m0Z0Z0 0Z0Z0Z 0Z0Z0Z0Z Z0Z0Z0 Z0Z0Z0Z0 0Z0Z0Z 0Z0Z0Z0Z m0Z0Z0 Z0Z0Z0Z0 (a) A 6 × 6 chessboard. (b) An 8 × 8 chessboard. Figure 3.16: Initial positions of re-entrant knight’s tours. 0ZQZ0Z0Z Z0Z0Z0ZQ 0Z0L0Z0Z Z0Z0Z0L0 0L0Z QZ0Z0Z0Z Z0ZQ Z0Z0ZQZ0 QZ0Z 0L0Z0Z0Z Z0L0 Z0Z0L0Z0 (a) n = 4 (b) n = 8 Figure 3.17: Solutions of the n-queens problem for n = 4, 8. 3.8. Problems 147 solve the n-queens problem for the case where n > 3. See [20] for a survey of the n-queens problem and its solutions. 3.20. Hampton Court Palace in England is well-known for its maze of hedges. Figure 3.18 shows a maze and its graph representation; the figure is adapted from page 434 in [169]. To obtain the graph representation, we use a vertex to represent an intersection in the maze. An edge joining two vertices represents a path from one intersection to another. (a) Suppose the entrance to the maze is represented by the lower-left black-filled vertex in Figure 3.18(b) and the exit is the upper-right black-filled vertex. Solve the maze by providing a path from the entrance to the exit. (b) Repeat the previous exercise for each pair of distinct vertices, letting one vertex of the pair be the entrance and the other vertex the exit. (c) What is the diameter of the graph in Figure 3.18(b)? (d) Investigate algorithms for generating and solving mazes. (a) (b) Figure 3.18: A maze and its graph representation. 3.21. For each of the algorithms below: (i) justify whether or not it can be applied to multigraphs or multidigraphs; (ii) if not, modify the algorithm so that it is applicable to multigraphs or multidigraphs. (a) Breadth-first search Algorithm 3.1. (b) Depth-first search Algorithm 3.2. (c) Graph connectivity test Algorithm 3.3. (d) General shortest path Algorithm 3.4. (e) Dijkstra’s Algorithm 3.5. (f) The Bellman-Ford Algorithms 3.6 and 3.7. (g) The Floyd-Roy-Warshall Algorithm 3.8. (h) The transitive closure Algorithm 3.9. (i) Johnson’s Algorithm 3.10. 148 Chapter 3. Shortest paths algorithms (a) 2 × 2 (b) 3 × 3 (c) 4 × 4 Figure 3.19: Grid graphs for n = 2, 3, 4. 3.22. Let n be a positive integer. An n × n grid graph is a graph on the Euclidean plane, where each vertex is an ordered pair from Z × Z. In particular, the vertices are ordered pairs (i, j) ∈ Z × Z such that 0 ≤ i, j < n. (3.12) Each vertex (i, j) is adjacent to any of the following vertices provided that ex- pression (3.12) is satisfied: the vertex (i − 1, j) immediately to its left, the vertex (i + 1, j) immediately to its right, the vertex (i, j + 1) immediately above it, or the vertex (i, j − 1) immediately below it. Figure 3.19 illustrates some examples of grid graphs. The 1 × 1 grid graph is the trivial graph K1 . (a) Fix a positive integer n > 1. Describe and provide pseudocode of an algorithm to generate all nonisomorphic n × n grid graphs. What is the worst-case runtime of your algorithm? (b) How many n × n grid graphs are there? How many of those graphs are nonisomorphic to each other? (c) Describe and provide pseudocode of an algorithm to generate a random n × n grid graph. Analyze the worst-case runtime of your algorithm. (d) Extend the grid graph by allowing edges to be diagonals. That is, a vertex (i, j) can also be adjacent to any of the following vertices so long as expres- sion (3.12) holds: (i − 1, j − 1), (i − 1, j + 1), (i + 1, j + 1), (i + 1, j − 1). With this extension, repeat the previous exercises. 3.23. Let G = (V, E) be a digraph with integer weight function w : E → Z\{0}, where either w(e) > 0 or w(e) < 0 for each e ∈ E. Yamada and Kinoshita [199] provide a divide-and-conquer algorithm to enumerate all the negative cycles in G. Investi- gate the divide and conquer technique for algorithm design. Describe and provide pseudocode of the Yamada-Kinoshita algorithm. Analyze its runtime complexity and prove the correctness of the algorithm. Chapter 4 Graph data structures — Randall Munroe, xkcd, http://xkcd.com/835/ What is the next task to be completed? Our daily lives are littered with priorities. There are tasks that take higher priorities than everything else and should be completed as soon as possible. Other tasks are of lower priorities and can be done whenever we have time. If we assign a nonnegative whole number to each task that needs to be completed, the task with the highest priority has the lowest whole number assigned to it. As each new task comes along, it is assigned a numerical priority. At any point in time, we want to choose and complete the task with the highest priority. This chapter will show how a structure called a priority queue can be used to efficiently choose the next task of highest priority. The structure can efficiently handle from a few tasks up to millions of tasks. In Chapters 3 and 2, we discussed various algorithms that rely on priority queues as one of their fundamental data structures. Such algorithms include Dijkstra’s algorithm, Prim’s algorithm, and the algorithm for constructing Huffman trees. The runtime of any algorithm that uses priority queues crucially depends on an efficient implementation of the priority queue data structure. This chapter discusses the general priority queue data structure and various efficient implementations based on trees. Section 4.1 provides some theoretical underpinning of priority queues and considers a simple implementation 149 150 Chapter 4. Graph data structures of priority queues as sorted lists. Section 4.2 discusses how to use binary trees to realize an efficient implementation of priority queues called a binary heap. Although very useful in practice, binary heaps do not lend themselves to being merged in an efficient manner, a setback rectified in section 4.3 by a priority queue implementation called binomial heaps. As a further application of binary trees, section 4.4 discusses binary search trees as a general data structure for managing data in a sorted order. 4.1 Priority queues A priority queue is a queue data structure with various rules on how to access and manage elements of the queue. Recall from section 3.2.1 that an ordinary queue Q has the following basic functions for accessing and managing its elements: dequeue(Q) — Remove the front of Q. enqueue(Q, e) — Append the element e to the end of Q. If Q is now a priority queue, each element is associated with a key or priority p ∈ X from a totally ordered set X. A binary relation denoted by an infix operator, say “≤”, is defined on all elements of X such that the following properties hold for all a, b, c ∈ X: Totality: We have a ≤ b or b ≤ a. Antisymmetry: If a ≤ b and b ≤ a, then a = b. Transitivity: If a ≤ b and b ≤ c, then a ≤ c. If the above three properties hold for the relation “≤”, then we say that “≤” is a total order on X and that X is a totally ordered set. In all, if the key of each element of Q belongs to the same totally ordered set X, we use the total order defined on X to compare the keys of the queue elements. For example, the set Z of integers is totally ordered by the “less than or equal to” relation. If the key of each e ∈ Q is an element of Z, we use the latter relation to compare the keys of elements of Q. In the case of an ordinary queue, the key of each queue element is its position index. To extract from a priority queue Q an element of lowest priority, we need to define the notion of smallest priority or key. Let pi be the priority or key assigned to element ei of Q. Then pmin is the lowest key if pmin ≤ p for any element key p. The element with corresponding key pmin is the minimum priority element. Based upon the notion of key comparison, we define two operations on a priority queue: insert(Q, e, p) — Insert into Q the element e with key p. extractMin(Q) — Extract from Q an element having the smallest priority. An immediate application of priority queues is sorting a finite sequence of items. Suppose L is a finite list of n > 0 items on which a total order is defined. Let Q be an empty priority queue. In the first phase of the priority queue sorting algorithm, we extract each element e ∈ L from L and insert e into Q with key e itself. In other words, each element e is its own key. This first phase of the sorting algorithm requires n element extractions from L and n element insertions into Q. The second phase of the algorithm involves extracting elements from Q via the extractMin operation. Queue elements are extracted via extractMin and inserted back into L in the order in which they are extracted from Q. Algorithm 4.1 presents pseudocode of our discussion. The runtime of Algorithm 4.1 depends on how the priority queue Q is implemented. 4.2. Binary heaps 151 Algorithm 4.1 Sorting a sequence via priority queue. Input A finite list L of n > 0 elements on which a total order is defined. Output The same list L sorted by the total order defined on its elements. 1: Q ← [ ] 2: for i ← 1, 2, . . . , n do 3: e ← extractMin(L) 4: enqueue(Q, e) 5: return Q 4.1.1 Sequence implementation A simple way to implement a priority queue is to maintain a sorted sequence. Let e0 , e1 , . . . , en be a sequence of n + 1 elements with corresponding keys κ0 , κ1 , . . . , κn and suppose that the κi all belong to the same totally ordered set X having total order ≤. Using the total order, we assume that the κi are sorted as κ0 ≤ κ1 ≤ · · · ≤ κn and ei ≤ ej if and only if κi ≤ κj . Then we consider the queue Q = [e0 , e1 , . . . , en ] as a priority queue in which the head is always the minimum element and the tail is always the maximum element. Extracting the minimum element is simply a dequeue operation that can be accomplished in constant time O(1). However, inserting a new element into Q takes linear time. Let e be an element with corresponding key κ ∈ X. Inserting e into Q requires that we maintain elements of Q sorted according to the total order ≤. If Q is empty, we simply enqueue e into Q. Suppose now that Q is a nonempty priority queue. If κ ≤ κ0 , then e becomes the new head of Q. If κn ≤ κ, then e becomes the new tail of Q. Inserting a new head or tail into Q each requires constant time O(1). However, if κ1 ≤ κ ≤ κn−1 then we need to traverse Q starting from e1 , searching for a position at which to insert e. Let ei be the queue element at position i within Q. If κ ≤ κi then we insert e into Q at position i, thus moving ei to position i + 1. Otherwise we next consider ei+1 and repeat the above comparison process. By hypothesis, κ1 ≤ κ ≤ κn−1 and therefore inserting e into Q takes a worst-case runtime of O(n). 4.2 Binary heaps A sequence implementation of priority queues has the advantage of being simple to understand. Inserting an element into a sequence-based priority queue requires linear time, which can quickly become infeasible for queues containing hundreds of thousands or even millions of elements. Can we do any better? Rather than using a sorted sequence, we can use a binary tree to realize an implementation of priority queues that is much more efficient than a sequence-based implementation. In particular, we use a data structure called a binary heap, which allows for element insertion in logarithmic time. In [197], Williams introduced the heapsort algorithm and described how to implement a priority queue using a binary heap. A basic idea is to consider queue elements as internal vertices in a binary tree T , with external vertices or leaves being “place-holders”. The tree T satisfies two further properties: 152 Chapter 4. Graph data structures 1. A relational property specifying the relative ordering and placement of queue ele- ments. 2. A structural property that specifies the structure of T . The relational property of T can be expressed as follows: Definition 4.1. Heap-order property. Let T be a binary tree and let v be a vertex of T other than the root. If p is the parent of v and these vertices have corresponding keys κp and κv , respectively, then κp ≤ κv . The heap-order property is defined in terms of the total order used to compare the keys of the internal vertices. Taking the total order to be the ordinary “less than or equal to” relation, it follows from the heap-order property that the root of T is always the vertex with a minimum key. Similarly, if the total order is the usual “greater than or equal to” relation, then the root of T is always the vertex with a maximum key. In general, if ≤ is a total order defined on the keys of T and u and v are vertices of T , we say that u is less than or equal to v if and only if u ≤ v. Furthermore, u is said to be a minimum vertex of T if and only if u ≤ v for all vertices of T . From our discussion above, the root is always a minimum vertex of T and is said to be “at the top of the heap”, from which we derive the name “heap” for this data structure. Another consequence of the heap-order property becomes apparent when we trace out a path from the root of T to any internal vertex. Let r be the root of T and let v be any internal vertex of T . If r, v0 , v1 , . . . , vn , v is an r-v path with corresponding keys κr , κv0 , κv1 , . . . , κvn , κv then we have κr ≤ κv0 ≤ κv1 ≤ · · · ≤ κvn ≤ κv . In other words, the keys encountered on the path from r to v are arranged in nonde- creasing order. The structural property of T is used to enforce that T be of as small a height as possible. Before stating the structural property, we first define the level of a binary tree. Recall that the depth of a vertex in T is its distance from the root. Level i of a binary tree T refers to all vertices of T that have the same depth i. We are now ready to state the heap-structure property. Definition 4.2. Heap-structure property. Let T be a binary tree with height h. Then T satisfies the heap-structure property if T is nearly a complete binary tree. That is, level 0 ≤ i ≤ h − 1 has 2i vertices, whereas level h has ≤ 2h vertices. The vertices at level h are filled from left to right. If a binary tree T satisfies both the heap-order and heap-structure properties, then T is referred to as a binary heap. By insisting that T satisfy the heap-order property, we are able to determine the minimum vertex of T in constant time O(1). Requiring that T also satisfy the heap-structure property allows us to determine the last vertex of T . The last vertex of T is identified as the right-most internal vertex of T having the greatest depth. Figure 4.1 illustrates various examples of binary heaps. The heap- structure property together with Theorem 2.21 result in the following corollary on the height of a binary heap. 4.2. Binary heaps 153 0 2 3 4 6 8 10 (a) 0 2 3 6 4 8 10 17 13 19 24 23 (b) 1 3 2 6 5 8 10 13 17 (c) Figure 4.1: Examples of binary heaps with integer keys. 154 Chapter 4. Graph data structures Corollary 4.3. A binary heap T with n internal vertices has height h = lg(n + 1) . Proof. Level h − 1 has at least one internal vertex. Apply Theorem 2.21 to see that T has at least 2h−2+1 − 1 + 1 = 2h−1 internal vertices. On the other hand, level h − 1 has at most 2h−1 internal vertices. Another application of Theorem 2.21 shows that T has at most 2h−1+1 − 1 = 2h − 1 internal vertices. Thus n is bounded by 2h−1 ≤ n ≤ 2h − 1. Taking logarithms of each side in the latter bound results in lg(n + 1) ≤ h ≤ lg n + 1 and the corollary follows. 0 2 3 6 4 8 10 17 13 19 24 23 0 2 3 4 6 8 10 (a) (b) 1 3 2 6 5 8 10 13 17 (c) Figure 4.2: Sequence representations of various binary heaps. 4.2.1 Sequence representation Any binary heap can be represented as a binary tree. Each vertex in the tree must know about its parent and its two children. However, a more common approach is to represent a binary heap as a sequence such as a list, array, or vector. Let T be a binary heap consisting of n internal vertices and let L be a list of n elements. The root vertex is represented as the list element L[0]. For each index i, the children of L[i] are L[2i + 1] and L[2i + 2] and the parent of L[i] is i−1 L . 2 4.2. Binary heaps 155 With a sequence representation of a binary heap, each vertex needs not know about its parent and children. Such information can be obtained via simple arithmetic on sequence indices. For example, the binary heaps in Figure 4.1 can be represented as the corresponding lists in Figure 4.2. Note that it is not necessary to store the leaves of T in the sequence representation. 4.2.2 Insertion and sift-up We now consider the problem of inserting a vertex v into a binary heap T . If T is empty, inserting a vertex simply involves the creation of a new internal vertex. We let that new internal vertex be v and let its two children be leaves. The resulting binary heap augmented with v has exactly one internal vertex and satisfies both the heap-order and heap-structure properties, as shown in Figure 4.3. In other words, any binary heap with one internal vertex trivially satisfies the heap-order property. v (a) (b) Figure 4.3: Inserting a vertex into an empty binary heap. Let T now be a nonempty binary heap, i.e. T has at least one internal vertex, and suppose we want to insert into T an internal vertex v. We must identify the correct leaf of T at which to insert v. If the n internal vertices of T are r = v0 , v1 , . . . , vn−1 , then by the sequence representation of T we can identify the last internal vertex vn−1 in constant time. The correct leaf at which to insert v is the sequence element immediately following vn−1 , i.e. the element at position n in the sequence representation of T . We replace with v the leaf at position n in the sequence so that v now becomes the last vertex of T . The binary heap T augmented with the new last vertex v satisfies the heap-structure property, but may violate the heap-order property. To ensure that T satisfies the heap- order property, we perform an operation on T called sift-up that involves possibly moving v up through various levels of T . Let κv be the key of v and let κp(v) be the key of v’s parent. If the relation κp(v) ≤ κv holds, then T satisfies the heap-order property. Otherwise we swap v with its parent, effectively moving v up one level to be at the position previously occupied by its parent. The parent of v is moved down one level and now occupies the position where v was previously. With v in its new position, we perform the same key comparison process with v’s new parent. The key comparison and swapping continue until the heap-order property holds for T . In the worst case, v would become the new root of T after undergoing a number of swaps that is proportional to the height of T . Therefore, inserting a new internal vertex into T can be achieved in time O(lg n). Figure 4.4 illustrates the insertion of a new internal vertex into a nonempty binary heap and the resulting sift-up operation to maintain the heap-order property. Algorithm 4.2 presents pseudocode of our discussion for inserting a new internal vertex into a nonempty binary heap. The pseudocode is adapted from Howard [103], which provides a C implementation of binary heaps. 156 Chapter 4. Graph data structures 1 1 2 3 2 3 6 4 8 10 6 4 8 10 17 13 19 24 23 17 13 19 24 23 0 (a) (b) 1 1 2 3 2 3 6 4 8 10 6 4 0 10 17 13 19 24 23 0 17 13 19 24 23 8 (c) (d) 1 1 2 3 2 0 6 4 0 10 6 4 3 10 17 13 19 24 23 8 17 13 19 24 23 8 (e) (f) 1 0 2 0 2 1 6 4 3 10 6 4 3 10 17 13 19 24 23 8 17 13 19 24 23 8 (g) (h) Figure 4.4: Insert and sift-up in a binary heap. 4.2. Binary heaps 157 Algorithm 4.2 Inserting a new internal vertex into a binary heap. Input A nonempty binary heap T , in sequence representation, having n internal vertices. An element v that is to be inserted as a new internal vertex of T . Output The binary heap T augmented with the new internal vertex v. 1: i ← n 2: while i > 0 do 3: p ← b(i − 1)/2c 4: if κT [p] ≤ κv then 5: exit the loop 6: else 7: T [i] ← T [p] 8: i←p 9: T [i] ← v 10: return T 4.2.3 Deletion and sift-down The process for deleting the minimum vertex of a binary heap bears some resemblance to that of inserting a new internal vertex into the heap. Having removed the minimum vertex, we must then ensure that the resulting binary heap satisfies the heap-order property. Let T be a binary heap. By the heap-order property, the root of T has a key that is minimum among all keys of internal vertices in T . If the root r of T is the only internal vertex of T , i.e. T is the trivial binary heap, we simply remove r and T now becomes the empty binary heap or the trivial tree, for which the heap-order property vacuously holds. Figure 4.5 illustrates the case of removing the root of a binary heap having one internal vertex. r (a) (b) Figure 4.5: Deleting the root of a trivial binary heap. We now turn to the case where T has n > 1 internal vertices. Let r be the root of T and let v be the last internal vertex of T . Deleting r would disconnect T . So we instead replace the key and information at r with the key and other relevant information pertaining to v. The root r now has the key of the last internal vertex, and v becomes a leaf. At this point, T satisfies the heap-structure property but may violate the heap-order property. To restore the heap-order property, we perform an operation on T called sift- down that may possibly move r down through various levels of T . Let c(r) be the child of r with key that is minimum among all the children of r, and let κr and κc(r) be the keys of r and c(r), respectively. If κr ≤ κc(r) , then the heap-order property is satisfied. Otherwise we swap r with c(r), moving r down one level to the position previously occupied by c(r). Furthermore, c(r) is moved up one level to the position previously occupied by r. With r in its new position, we perform the same key comparison process with a child of r that has minimum key among all of r’s children. The key comparison and swapping 158 Chapter 4. Graph data structures continue until the heap-order property holds for T . In the worst case, r would percolate all the way down to the level that is immediately above the last level after undergoing a number of swaps that is proportional to the height of T . Therefore, deleting the minimum vertex of T can be achieved in time O(lg n). Figure 4.6 illustrates the deletion of the minimum vertex of a binary heap with at least two internal vertices and the resulting sift-down process that percolates vertices down through various levels of the heap in order to maintain the heap-order property. Algorithm 4.3 summarizes our discussion of the process for extracting the minimum vertex of T while also ensuring that T satisfies the heap-order property. The pseudocode is adapted from the C implementation of binary heaps in Howard [103]. With some minor changes, Algorithm 4.3 can be used to change the key of the root vertex and maintain the heap-order property for the resulting binary tree. Algorithm 4.3 Extract the minimum vertex of a binary heap. Input A binary heap T , given in sequence representation, having n > 1 internal vertices. Output Extract the minimum vertex of T . With one vertex removed, T must satisfy the heap-order property. 1: root ← T [0] 2: n ← n − 1 3: v ← T [n] 4: i ← 0 5: j ← 0 6: while true do 7: left ← 2i + 1 8: right ← 2i + 2 9: if left < n and κT [left] ≤ κv then 10: if right < n and κT [right] ≤ κT [left] then 11: j ← right 12: else 13: j ← left 14: else if right < n and κT [right] ≤ κv then 15: j ← right 16: else 17: T [i] ← v 18: exit the loop 19: T [i] ← T [j] 20: i←j 21: return root 4.2.4 Constructing a binary heap Given a collection of n vertices v0 , v1 , . . . , vn−1 with corresponding keys κ0 , κ1 , . . . , κn−1 , we want to construct a binary heap containing exactly those vertices. A basic approach is to start with a trivial tree and build up a binary heap via successive insertions. As each insertion requires O(lg n) time, the method of binary heap construction via successive insertion of each of the n vertices requires O(n · lg n) time. It turns out we could do a bit better and achieve the same result in linear time. 4.2. Binary heaps 159 1 23 2 3 2 3 6 4 8 10 6 4 8 10 17 13 19 24 23 17 13 19 24 (a) (b) 23 2 2 3 23 3 6 4 8 10 6 4 8 10 17 13 19 24 17 13 19 24 (c) (d) 2 2 23 3 4 3 6 4 8 10 6 23 8 10 17 13 19 24 17 13 19 24 (e) (f) 2 2 4 3 4 3 6 23 8 10 6 19 8 10 17 13 19 24 17 13 23 24 (g) (h) Figure 4.6: Delete and sift-down in a binary heap. 160 Chapter 4. Graph data structures Algorithm 4.4 Heapify a binary tree. Input A binary tree T , given in sequence representation, having n > 1 internal vertices. Output The binary tree T heapified so that it satisfies the heap-order property. 1: for i ← bn/2c − 1, . . . , 0 do 2: v ← T [i] 3: j←0 4: while true do 5: left ← 2i + 1 6: right ← 2i + 2 7: if left < n and κT [left] ≤ κv then 8: if right < n and κT [right] ≤ κT [left] then 9: j ← right 10: else 11: j ← left 12: else if right < n and κT [right] ≤ κv then 13: j ← right 14: else 15: T [i] ← v 16: exit the while loop 17: T [i] ← T [j] 18: i←j 19: return T A better approach starts by letting v0 , v1 , . . . , vn−1 be the internal vertices of a binary tree T . The tree T need not satisfy the heap-order property, but it must satisfy the heap- structure property. Suppose T is given in sequence representation so that we have the correspondence vi = T [i] and the last internal vertex of T has index n − 1. The parent of T [n − 1] has index n−1 j= . 2 Any vertex of T with sequence index beyond n − 1 is a leaf. In other words, if an internal vertex has index > j, then the children of that vertex are leaves and have indices ≥ n. Thus any internal vertex with index ≥ bn/2c has leaves for its children. Conclude that internal vertices with indices jnk jnk jnk , + 1, + 2, . . . , n − 1 (4.1) 2 2 2 have only leaves for their children. Our next task is to ensure that the heap-order property holds for T . If v is an internal vertex with index in (4.1), then the subtree rooted at v is trivially a binary heap. Consider the indices from bn/2c − 1 all the way down to 0 and let i be such an index, i.e. let 0 ≤ i ≤ bn/2c − 1. We heapify the subtree of T rooted at T [i], effectively performing a sift-down on this subtree. Once we have heapified all subtrees rooted at T [i] for 0 ≤ i ≤ bn/2c − 1, the resulting tree T is a binary heap. Our discussion is summarized in Algorithm 4.4. Earlier in this section, we claimed that Algorithm 4.4 can be used to construct a binary heap in worst-case linear time. To prove this, let T be a binary tree satisfying the 4.3. Binomial heaps 161 heap-structure property and having n internal vertices. By Corollary 4.3, T has height h = dlg(n + 1)e. We perform a sift-down for at most 2i vertices of depth i, where each sift-down for a subtree rooted at a vertex of depth i takes O(h − i) time. Then the total time for Algorithm 4.4 is ! ! X X 2−i i h O 2 (h − i) = O 2 0≤i<h 0≤i<h 2h−i ! X k = O 2h k>0 2k = O 2h+1 = O(n) P where we used the closed form k>0 k/2k = 2 for a geometric series and Theorem 2.21. 4.3 Binomial heaps We are given two binary heaps T1 and T2 and we want to merge them into a single heap. We could start by choosing to insert each element of T2 into T1 , successively extracting the minimum element from T2 and insert that minimum element into T1 . If T1 and T2 have m and n elements, respectively, we would perform n extractions from T2 totalling ! X O lg k 0<k≤n time and inserting all of the extracted elements from T2 into T1 requires a total runtime of ! X O lg k . (4.2) n≤k<n+m We approximate the addition of the two sums by Z n+m k=n+m k ln k − k lg k dk = +C 0 ln 2 k=0 for some constant C. The above method of successive extraction and insertion therefore has a total runtime of (n + m) ln(n + m) − n − m O ln 2 for merging two binary heaps. Alternatively, we could slightly improve the latter runtime for merging T1 and T2 by successively extracting the last internal vertex of T2 . The whole process of extracting all elements from T2 in this way takes O(n) time and inserting each of the extracted elements into T1 still requires the runtime in expression (4.2). We approximate the sum in (4.2) by Z k=n+m k=n+m k ln k − k lg k dk = +C k=n ln 2 k=n 162 Chapter 4. Graph data structures for some constant C. Therefore the improved extraction and insertion method requires (n + m) ln(n + m) − n ln n − m O −n ln 2 time in order to merge T1 and T2 . Can we improve on the latter runtime for merging two binary heaps? It turns out we can by using a type of mergeable heap called binomial heap that supports merging two heaps in logarithmic time. 4.3.1 Binomial trees A binomial heap can be considered as a collection of binomial trees. The binomial tree of order k is denoted Bk and defined recursively as follows: 1. The binomial tree of order 0 is the trivial tree. 2. The binomial tree of order k > 0 is a rooted tree, where from left to right the children of the root of Bk are roots of Bk−1 , Bk−2 , . . . , B0 . Various examples of binomial trees are shown in Figure 4.7. The binomial tree Bk can also be defined as follows. Let T1 and T2 be two copies of Bk−1 with root vertices r1 and r2 , respectively. Then Bk is obtained by letting, say, r1 be the left-most child of r2 . Lemma 4.4 lists various basic properties of binomial trees. Property (3) of Lemma 4.4 uses the binomial coefficient, from whence Bk derives its name. Lemma 4.4. Basic properties of binomial trees. Let Bk be a binomial tree of order k ≥ 0. Then the following properties hold: 1. The order of Bk is 2k . 2. The height of Bk is k. k 3. For 0 ≤ i ≤ k, we have i vertices at depth i. 4. The root of Bk is the only vertex with maximum degree ∆(Bk ) = k. If the children of the root are numbered k − 1, k − 2, . . . , 0 from left to right, then child i is the root of the subtree Bi . Proof. We use induction on k. The base case for each of the above properties is B0 , which trivially holds. (1) By our inductive hypothesis, Bk−1 has order 2k−1 . Since Bk is comprised of two copies of Bk−1 , conclude that Bk has order 2k−1 + 2k−1 = 2k . (2) The binomial tree Bk is comprised of two copies of Bk−1 , the root of one copy being the left-most child of the root of the other copy. Then the height of Bk is one greater than the height of Bk−1 . By our inductive hypothesis, Bk−1 has height k − 1 and therefore Bk has height (k − 1) + 1 = k. 4.3. Binomial heaps 163 (a) B0 (b) B1 (c) B2 (d) B3 (e) B4 (f) B5 Figure 4.7: Binomial trees Bk for k = 0, 1, 2, 3, 4, 5. 164 Chapter 4. Graph data structures (3) Denote by D(k, i) the number of vertices of depth i in Bk . As Bk is comprised of two copies of Bk−1 , a vertex at depth i in Bk−1 appears once in Bk at depth i and a second time at depth i + 1. By our inductive hypothesis, D(k, i) = D(k − 1, i) + D(k − 1, i − 1) k−1 k−1 = + i i−1 k = i where we used Pascal’s formula which states that n+1 n n = + r r−1 r for any positive integers n and r with r ≤ n. (4) This property follows from the definition of Bk . Corollary 4.5. If a binomial tree has order n ≥ 0, then the degree of any vertex i is bounded by deg(i) ≤ lg n. Proof. Apply properties (1) and (4) of Lemma 4.4. 4.3.2 Binomial heaps In 1978, Jean Vuillemin [188] introduced binomial heaps as a data structure for im- plementing priority queues. Mark R. Brown [41, 42] subsequently extended Vuillemin’s work, providing detailed analysis of binomial heaps and introducing an efficient imple- mentation. A binomial heap H can be considered as a collection of binomial trees. Each vertex in H has a corresponding key and all vertex keys of H belong to a totally ordered set having total order ≤. The heap also satisfies the following binomial heap properties: Heap-order property. Let Bk be a binomial tree in H. If v is a vertex of Bk other than the root and p is the parent of v and having corresponding keys κv and κp , respectively, then κp ≤ κv . Root-degree property. For any integer k ≥ 0, H contains at most one binomial tree whose root has degree k. If H is comprised of the binomial trees Bk0 , Bk1 , . . . , Bkn for nonnegative integers ki , we can consider H as a forest made up of the trees Bki . We can also represent H as a tree in the following way. List the binomial trees of H as Bk0 , Bk1 , . . . , Bkn in nondecreasing order of root degrees, i.e. the root of Bki has order less than or equal to the root of Bkj if and only if ki ≤ kj . The root of H is the root of Bk0 and the root of each Bki has for its child the root of Bki+1 . Both the forest and tree representations are illustrated in Figure 4.8 for the binomial heap comprised of the binomial trees B0 , B1 , B3 . The heap-order property for binomial heaps is analogous to the heap-order property for binary heaps. In the case of binomial heaps, the heap-order property implies that the root of a binomial tree has a key that is minimum among all vertices in that tree. 4.3. Binomial heaps 165 (a) Binomial heap as a forest. (b) Binomial heap as a tree. Figure 4.8: Forest and tree representations of a binomial heap. However, the similarity more or less ends there. In a tree representation of a binomial heap, the root of the heap may not necessarily have the minimum key among all vertices of the heap. The root-degree property can be used to derive an upper bound on the number of binomial trees in a binomial heap. If H is a binomial heap with n vertices, then H has at most 1 + blg nc binomial trees. To prove this result, note that (see Theorem 2.1 and Corollary 2.1.1 in [166, pp.40–42]) n can be uniquely written in binary representation as the polynomial n = ak 2k + ak−1 2k−1 + · · · + a1 21 + a0 20 . Pblg nc i The binary representation of n requires 1 + blg nc bits, hence n = i=0 ai 2 . Apply property (1) of Lemma 4.4 to see that the binomial tree Bi is in H if and only if the i-th bit is bi = 1. Conclude that H has at most 1 + blg nc binomial trees. 4.3.3 Construction and management Let H be a binomial heap comprised of the binomial trees Bk0 , Bk1 , . . . , Bkn where the root of Bki has order less than or equal to the root of Bkj if and only if ki ≤ kj . Denote by rki the root of the binomial tree Bki . If v is a vertex of H, denote by child[v] the left-most child of v and by sibling[v] we mean the sibling immediately to the right of v. Furthermore, let parent[v] be the parent of v and let degree[v] denote the degree of v. If v has no children, we set child[v] = NULL. If v is one of the roots rki , we set parent[v] = NULL. And if v is the right-most child of its parent, then we set sibling[v] = NULL. The roots rk0 , rk1 , . . . , rkn can be organized as a linked list, called a root list, with two functions for accessing the next root and the previous root. The root immediately following rki is denoted next[rki ] = sibling[v] = rki+1 and the root immediately before rki is written prev[rki ] = rki−1 . For rk0 and rkn , we set next[rkn ] = sibling[v] = NULL and prev[rk0 ] = NULL. We also define the function head[H] that simply returns rk0 whenever H has at least one element, and head[H] = NULL otherwise. Minimum vertex To find the minimum vertex, we find the minimum among rk0 , rk1 , . . . , rkm because by definition the root rki is the minimum vertex of the binomial tree Bki . If H has n vertices, 166 Chapter 4. Graph data structures we need to check at most 1 + blg nc vertices to find the minimum vertex of H. Therefore determining the minimum vertex of H takes O(lg n) time. Algorithm 4.5 summarizes our discussion. Algorithm 4.5 Determine the minimum vertex of a binomial heap. Input A binomial heap H of order n > 0. Output The minimum vertex of H. 1: u ← Null 2: v ← head[H] 3: min ← ∞ 4: while v 6= Null do 5: if κv < min then 6: min ← κv 7: u←v 8: v ← sibling[v] 9: return u Merging heaps Recall that Bk is constructed by linking the root of one copy of Bk−1 with the root of another copy of Bk−1 . When merging two binomial heaps whose roots have the same degree, we need to repeatedly link the respective roots. The root linking procedure runs in constant time O(1) and is rather straightforward, as presented in Algorithm 4.6. Algorithm 4.6 Linking the roots of binomial heaps. Input Two copies of Bk−1 , one rooted at u and the other at v. Output The respective roots of two copies of Bk−1 linked, with one root becoming the parent of the other. 1: parent[u] ← v 2: sibling[u] ← child[v] 3: child[v] ← u 4: degree[v] ← degree[v] + 1 Besides linking the roots of two copies of Bk−1 , we also need to merge the root lists of two binomial heaps H1 and H2 . The resulting merged list is sorted in nondecreasing order of degree. Let L1 be the root list of H1 and let L2 be the root list of H2 . First we create an empty list L. As the lists Li are already sorted in nondecreasing order of vertex degree, we use merge sort to merge the Li into a single sorted list. The whole procedure for merging the Li takes linear time O(n), where n = |L1 | + |L2 | − 1. Refer to Algorithm 4.7 for pseudocode of the procedure just described. Having clarified the root linking and root lists merging procedures, we are now ready to describe a procedure for merging two nonempty binomial heaps H1 and H2 into a single binomial heap H. Initially there are at most two copies of B0 , one from each of the Hi . If two copies of B0 are present, we let the root of one be the parent of the other as per Algorithm 4.6, producing B1 as a result. From thereon, we generally have at most three copies of Bk for some integer k > 0: one from H1 , one from H2 , and the third from a previous merge of two copies of Bk−1 . In the presence of two or more copies of Bk , we 4.3. Binomial heaps 167 Algorithm 4.7 Merging two root lists. Input Two root lists L1 and L2 , each containing the roots of binomial trees in the binomial heaps H1 and H2 , respectively. Each root list Li is sorted in increasing order of vertex degree. Output A single list L that merges the root lists Li and sorted in nondecreasing order of degree. 1: i ← 1 2: j ← 1 3: L ← [ ] 4: n ← |L1 | + |L2 | − 1 5: append(L1 , ∞) 6: append(L2 , ∞) 7: for k ← 0, 1, . . . , n do 8: if deg(L1 [i]) ≤ deg(L2 [j]) then 9: append(L, L1 [i]) 10: i←i+1 11: else 12: append(L, L2 [j]) 13: j ←j+1 14: return L merge two copies as per Algorithm 4.6 to produce Bk+1 . If Hi has ni vertices, then Hi has at most 1 + blg ni c binomial trees, from which it is clear that merging H1 and H2 requires max(1 + blg n1 c, 1 + blg n2 c) steps. Letting N = max(n1 , n2 ), we see that merging H1 and H2 takes logarithmic time O(lg N ). The operation of merging two binomial heaps is presented in pseudocode as Algorithm 4.8, which is adapted from Cormen et al. [57, p.463] and the C implementation of binomial queues in [103]. A word of warning is order here. Algorithm 4.8 is destructive in the sense that it modifies the input heaps Hi in-place without making copies of those heaps. Vertex insertion Let v be a vertex with corresponding key κv and let H1 be a binomial heap of n vertices. The single vertex v can be considered as a binomial heap H2 comprised of exactly the binomial tree B0 . Then inserting v into H1 is equivalent to merging the heaps Hi and can be accomplished in O(lg n) time. Refer to Algorithm 4.9 for pseudocode of this straightforward procedure. Delete minimum vertex Extracting the minimum vertex from a binomial heap H consists of several phases. Let H be comprised of the binomial trees Bk0 , Bk1 , . . . , Bkm with corresponding roots rk0 , rk1 , . . . , rkm and let n be the number of vertices in H. In the first phase, from among the rki we identify the root v with minimum key and remove v from H, an operation that runs in O(lg n) time because we need to process at most 1 + blg nc roots. With the binomial tree Bk rooted at v thus severed from H, we now have a forest consisting of the 168 Chapter 4. Graph data structures Algorithm 4.8 Merging two binomial heaps. Input Two binomial heaps H1 and H2 . Output A binomial heap H that results from merging the Hi . 1: H ← empty binomial heap 2: head[H] ← merge sort the root lists of H1 and H2 3: if head[H] = Null then 4: return H 5: prevv ← Null 6: v ← head[H] 7: nextv ← sibling[v] 8: while nextv 6= Null do 9: if degree[v] 6= degree[nextv] or (sibling[nextv] 6= Null and degree[sibling[nextv]] = degree[v]) then 10: prevv ← v 11: v ← nextv 12: else if κv ≤ κnextv then 13: sibling[v] ← sibling[nextv] 14: link the roots nextv and v as per Algorithm ?? 15: else 16: if prevv = Null then 17: head[H] ← nextv 18: else 19: sibling[prevv] ← nextv 20: link the roots v and nextv as per Algorithm ?? 21: v ← nextv 22: nextv ← sibling[v] 23: return H Algorithm 4.9 Insert a vertex into a binomial heap. Input A binomial heap H and a vertex v. Output The heap H with v inserted into it. 1: H1 ← empty binomial heap 2: head[H1 ] ← v 3: parent[v] ← Null 4: child[v] ← Null 5: sibling[v] ← Null 6: degree[v] ← 0 7: H ← merge H and H1 as per Algorithm ?? 4.4. Binary search trees 169 heap without Bk (denote this heap by H1 ) and the binomial tree Bk . By construction, v is the root of Bk and the children of v from left to right can be considered as roots of binomial trees as well, say B`s , B`s−1 , . . . , B`0 where `s > `s−1 > · · · > `0 . Now sever the root v from its children. The B`j together can be viewed as a binomial heap H2 with, from left to right, binomial trees B`0 , B`1 , . . . , B`s . Finally the binomial heap resulting from removing v can be obtained by merging H1 and H2 in O(lg n) time as per Algorithm 4.8. In total we can extract the minimum vertex of H in O(lg n) time. Our discussion is summarized in Algorithm 4.10 and an illustration of the extraction process is presented in Figure 4.9. Algorithm 4.10 Extract the minimum vertex from a binomial heap. Input A binomial heap H. Output The minimum vertex of H removed. 1: v ← extract minimum vertex from root list of H 2: H2 ← empty binomial heap 3: L ← list of v’s children reversed 4: head[H2 ] ← L[0] 5: H ← merge H and H2 as per Algorithm ?? 6: return v 4.4 Binary search trees A binary search tree (BST) is a rooted binary tree T = (V, E) having vertex weight function κ : V → R. The weight of each vertex v is referred to as its key, denoted κv . Each vertex v of T satisfies the following properties: Left subtree property. The left subtree of v contains only vertices whose keys are at most κv . That is, if u is a vertex in the left subtree of v, then κu ≤ κv . Right subtree property. The right subtree of v contains only vertices whose keys are at least κv . In other words, any vertex u in the right subtree of v satisfies κv ≤ κu . Recursion property. Both the left and right subtrees of v must also be binary search trees. The above are collectively called the binary search tree property. See Figure 4.10 for an example of a binary search tree. Based on the binary search tree property, we can use in-order traversal (see Algorithm 2.12) to obtain a listing of the vertices of a binary search tree sorted in nondecreasing order of keys. 4.4.1 Searching Given a BST T and a key k, we want to locate a vertex (if one exists) in T whose key is k. The search procedure for a BST is reminiscent of the binary search algorithm discussed in problem 3.8. We begin by examining the root v0 of T . If κv0 = k, the search is successful. However, if κv0 6= k then we have two cases to consider. In the first case, if k < κv0 then we search the left subtree of v0 . The second case occurs when k > κv0 , 170 Chapter 4. Graph data structures 70 65 60 40 1 67 66 68 41 43 44 2 3 5 7 69 45 48 49 3 6 9 8 12 15 47 4 8 11 9 10 (a) 70 65 60 40 1 67 66 68 41 43 44 2 3 5 7 69 45 48 49 3 6 9 8 12 15 47 4 8 11 9 10 (b) 70 65 60 40 7 5 3 2 67 66 68 41 43 44 15 8 12 3 6 9 69 45 48 49 9 4 8 11 47 10 (c) 7 5 3 2 70 65 15 60 8 12 40 3 6 9 67 66 68 9 41 43 44 4 8 11 69 45 48 49 10 47 (d) Figure 4.9: Extracting the minimum vertex from a binomial heap. 4.4. Binary search trees 171 10 5 15 4 7 13 20 3 5 5 8 11 Figure 4.10: A binary search tree. in which case we search the right subtree of v0 . Repeat the process until a vertex v in T is found for which k = κv or the indicated subtree is empty. Whenever the target key is different from the key of the vertex we are currently considering, we move down one level of T . Thus if h is the height of T , it follows that searching T takes a worst-case runtime of O(h). The above procedure is presented in pseudocode as Algorithm 4.11. Note that if a vertex v does not have a left subtree, the operation of locating the root of v’s left subtree should return NULL. A similar comment applies when v does not have a right subtree. Furthermore, from the structure of Algorithm 4.11, if the input BST is empty then NULL is returned. See Figure 4.11 for an illustration of locating vertices with given keys in a BST. Algorithm 4.11 Locate a key in a binary search tree. Input A binary search tree T and a target key k. Output A vertex in T with key k. If no such vertex exists, return Null. 1: v ← root[T ] 2: while v 6= Null and k 6= κv do 3: if k < κv then 4: v ← leftchild[v] 5: else 6: v ← rightchild[v] 7: return v 10 10 5 15 5 15 4 7 13 20 4 7 13 20 3 5 5 8 11 19 23 3 5 5 8 11 19 23 2 3 9 22 26 2 3 9 22 26 (a) Vertex with key 6: search fail. (b) Vertex with key 22: search success. Figure 4.11: Finding vertices with given keys in a BST. From the binary search tree property, deduce that a vertex of a BST T with minimum key can be found by starting from the root of T and repeatedly traversing left subtrees. 172 Chapter 4. Graph data structures 10 10 5 15 5 15 4 7 13 20 4 7 13 20 3 5 5 8 11 19 23 3 5 5 8 11 19 23 2 3 9 22 26 2 3 9 22 26 (a) Minimum vertex. (b) Maximum vertex. Figure 4.12: Locating minimum and maximum vertices in a BST. 10 10 5 15 5 15 4 7 13 20 4 7 13 20 3 5 5 8 11 19 23 3 5 5 8 11 19 23 2 3 9 22 26 2 3 9 22 26 (a) Successor of 9. (b) Predecessor of 11. Figure 4.13: Searching for successor and predecessor. 4.4. Binary search trees 173 When we have reached the left-most vertex v of T , querying for the left subtree of v should return NULL. At this point, we conclude that v is a vertex with minimum key. Each query for the left subtree moves us one level down T , resulting in a worst-case runtime of O(h) with h being the height of T . See Algorithm 4.12 for pseudocode of the procedure. The procedure for finding a vertex with maximum key is analogous to that for finding one with minimum key. Starting from the root of T , we repeatedly traverse right subtrees until we encounter the right-most vertex, which by the binary search tree property has maximum key. This procedure has the same worst-case runtime of O(h). Figure 4.12 illustrates the process of locating the minimum and maximum vertices of a BST. Algorithm 4.12 Finding a vertex with minimum key in a BST. Input A nonempty binary search tree T . Output A vertex of T with minimum key. 1: v ← root of T 2: while leftchild[v] 6= Null do 3: v ← leftchild[v] 4: return v Corresponding to the notions of left- and right-children, we can also define successors and predecessors as follows. Suppose v is not a maximum vertex of a nonempty BST T . The successor of v is a vertex in T distinct from v with the smallest key greater than or equal to κv . Similarly, for a vertex v that is not a minimum vertex of T , the predecessor of v is a vertex in T distinct from v with the greatest key less than or equal to κv . The notions of successors and predecessors are concerned with relative key order, not a vertex’s position within the hierarchical structure of a BST. For instance, from Figure 4.10 we see that the successor of the vertex u with key 8 is the vertex v with key 10, i.e. the root, even though v is an ancestor of u. The predecessor of the vertex a with key 4 is the vertex b with key 3, i.e. the minimum vertex, even though b is a descendant of a. We now describe a method to systematically locate the successor of a given vertex. Let T be a nonempty BST and v ∈ V (T ) not a maximum vertex of T . If v has a right subtree, then we find a minimum vertex of v’s right subtree. In case v does not have a right subtree, we backtrack up one level to v’s parent u = parent(v). If v is the root of the right subtree of u, we backtrack up one level again to u’s parent, making the assignments v ← u and u ← parent(u). Otherwise we return v’s parent. Repeat the above backtracking procedure until the required successor is found. Our discussion is summarized in Algorithm 4.13. Each time we backtrack to a vertex’s parent, we move up one level, hence the worst-case runtime of Algorithm 4.13 is O(h) with h being the height of T . The procedure for finding predecessors is similar. Refer to Figure 4.13 for an illustration of locating successors and predecessors. 4.4.2 Insertion Inserting a vertex v into a BST T is rather straightforward. If T is empty, we let v be the root of T . Otherwise T has at least one vertex. In that case, we need to locate a vertex in T that can act as a parent and “adopt” v as a child. To find a candidate parent, let u be the root of T . If κv < κu then we assign the root of the left subtree of u to u itself. 174 Chapter 4. Graph data structures Algorithm 4.13 Finding successors in a binary search tree. Input A nonempty binary search tree T and a vertex v that is not a maximum of T . Output The successor of v. 1: if rightchild[v] 6= Null then 2: return minimum vertex of v’s right subtree as per Algorithm ?? 3: u ← parent(v) 4: while u 6= Null and v = rightchild[u] do 5: v←u 6: u ← parent(u) 7: return u Otherwise we assign the root of the right subtree of u to u. We then carry on the above key comparison process until the operation of locating the root of a left or right subtree returns NULL. At this point, a candidate parent for v is the last non-NULL value of u. If κv < κu then we let v be u’s left-child. Otherwise v is the right-child of u. After each key comparison, we move down at most one level so that in the worst-case inserting a vertex into T takes O(h) time, where h is the height of T . Algorithm 4.14 presents pseudocode of our discussion and Figure 4.14 illustrates how to insert a vertex into a BST. Algorithm 4.14 Inserting a vertex into a binary search tree. Input A binary search tree T and a vertex x to be inserted into T . Output The same BST T but augmeneted with x. 1: u ← Null 2: v ← root of T 3: while v 6= Null do 4: u←v 5: if κx < κv then 6: v ← leftchild[v] 7: else 8: v ← rightchild[v] 9: parent[x] ← u 10: if u = Null then 11: root[T ] ← x 12: else 13: if κx < κu then 14: leftchild[u] ← x 15: else 16: rightchild[u] ← x 4.4.3 Deletion Whereas insertion into a BST is straightforward, removing a vertex requires much more work. Let T be a nonempty binary search tree and suppose we want to remove v ∈ V (T ) from T . Having located the position that v occupies within T , we need to consider three separate cases: (1) v is a leaf; (2) v has one child; (3) v has two children. 4.4. Binary search trees 175 10 10 5 15 5 15 4 7 13 20 4 7 13 20 3 5 5 8 11 19 23 3 5 5 8 11 19 23 2 3 9 22 26 2 3 9 12 22 26 (a) (b) Figure 4.14: Inserting into a binary search tree. Algorithm 4.15 Deleting a vertex from a binary search tree. Input A nonempty binary search tree T and a vertex x ∈ V (T ) to be removed from T . Output The same BST T but without x. 1: u ← Null 2: v ← Null 3: if leftchild[x] 6= Null or rightchild[x] 6= Null then 4: v←x 5: else 6: v ← successor of x 7: if leftchild[v] 6= Null then 8: u ← leftchild[v] 9: else 10: u ← rightchild[v] 11: if u 6= Null then 12: parent[u] ← parent[v] 13: if parent[v] = Null then 14: root[T ] ← u 15: else 16: if v = leftchild[parent[v]] then 17: leftchild[parent[v]] ← u 18: else 19: rightchild[parent[v]] ← u 20: if v 6= x then 21: κx ← κv 22: copy v’s auxilary data into x 176 Chapter 4. Graph data structures 1. If v is a leaf, we simply remove v from T and the procedure is complete. The resulting tree without v satisfies the binary search tree property. 2. Suppose v has the single child u. Removing v would disconnect T , a situation that can be prevented by splicing out u and letting u occupy the position previously held by v. The resulting tree with v removed as described satisfies the binary search tree property. 3. Finally suppose v has two children and let s and p be the successor and predecessor of v, respectively. It can be shown that s has no left-child and p has no right-child. We can choose to either splice out s or p. Say we choose to splice out s. Then we remove v and let s hold the position previously occupied by v. The resulting tree with v thus removed satisfies the binary search tree property. The above procedure is summarized in Algorithm 4.15, which is adapted from [57, p.262]. Figure 4.15 illustrates the various cases to be considered when removing a vertex from a BST. Note that in Algorithm 4.15, the process of finding the successor dominates the runtime of the entire algorithm. Other operations in the algorithm take at most constant time. Therefore deleting a vertex from a binary search tree can be accomplished in worst- case O(h) time, where h is the height of the BST under consideration. 4.5 Problems No problem is so formidable that you can’t walk away from it. — Charles M. Schulz 4.1. Let Q be a priority queue of n > 1 elements, given in sequence representation. From section 4.1.1, we know that inserting an element into Q takes O(n) time and deleting an element from Q takes O(1) time. (a) Suppose Q is an empty priority queue and let e0 , e1 , . . . , en be n + 1 elements we want to insert into Q. What is the total runtime required to insert all the ei into Q while also ensuring that the resulting queue is a priority queue? (b) Let Q = [e0 , e1 , . . . , en ] be a priority queue of n + 1 elements. What is the total time required to remove all the elements of Q? 4.2. Prove the correctness of Algorithms 4.2 and 4.3. 4.3. Describe a variant of Algorithm 4.3 for modifying the key of the root of a binary heap, without extracting any vertex from the heap. 4.4. Section 4.2.2 describes how to insert an element into a binary heap T . The general strategy is to choose the first leaf following the last internal vertex of T , replace that leaf with the new element so that it becomes an internal vertex, and perform a sift-up operation from there. If instead we choose any leaf of T and replace that leaf with the new element, explain why we cannot do any better than Algorithm 4.2. 4.5. Section 4.2.3 shows how to extract the minimum vertex from a binary heap T . Instead of replacing the root with the last internal vertex of T , we could replace the root with any other vertex of T that is not a leaf and then proceed to maintain the heap-structure and heap-order properties. Explain why the latter strategy is not better than Algorithm 4.3. 4.5. Problems 177 10 10 5 15 5 15 4 7 13 20 4 7 13 20 3 5 5 8 11 19 23 3 5 5 8 11 19 23 2 3 9 22 26 2 3 22 26 (a) Target vertex 9 is a leaf. (b) Leaf deleted. 10 10 5 15 5 15 4 7 13 20 4 7 12 20 3 5 5 8 12 19 23 3 5 5 8 11 19 23 2 3 11 22 26 2 3 22 26 (c) Target vertex 13 has one child. (d) Vertex deleted. 10 5 15 10 4 7 13 20 5 16 3 5 5 8 12 18 23 4 7 13 20 2 3 11 17 19 22 26 3 5 5 8 12 18 23 16 2 3 11 17 19 22 26 (e) Target vertex 15 has two children. (f) Vertex deleted. Figure 4.15: Deleting a vertex from a binary search tree. 178 Chapter 4. Graph data structures 20 4 15 5 13 7 13 10 10 10 7 15 7 13 5 15 5 10 20 5 15 4 7 13 20 4 4 20 (a) (b) (c) (d) Figure 4.16: Different structural representations of a BST. 4.6. Let S be a sequence of n > 1 real numbers. How can we use algorithms described in section 4.2 to sort S? 4.7. The binary heaps discussed in section 4.2 are properly called minimum binary heaps because the root of the heap is always the minimum vertex. A correspond- ing notion is that of maximum binary heaps, where the root is always the maximum element. Describe algorithms analogous to those in section 4.2 for managing max- imum binary heaps. 4.8. What is the total time required to extract all elements from a binary heap? 4.9. Numbers of the form nr are called binomial coefficients. They also count the number of r-combinations from a set of n objects. Algorithm ?? presents pseu- docode to generate all the r-combinations of a set of n distinct objects. What is the worst-case runtime of Algorithm ??? Prove the correctness of Algorithm ??. 4.10. In contrast to enumerating all the r-combinations of a set of n objects, we may only want to generate a random r-combination. Describe and present pseudocode of a procedure to generate a random r-combination of {1, 2, . . . , n}. 4.11. A problem related to the r-combinations of the set S = {1, 2, . . . , n} is that of generating the permutations of S. Algorithm ?? presents pseudocode to generate all the permutations of S in increasing lexicographic order. Find the worst-case runtime of this algorithm and prove its correctness. 4.12. Provide a description and pseudocode of an algorithm to generate a random per- mutation of {1, 2, . . . , n}. 4.13. Takaoka [179] presents a general method for combinatorial generation that runs in O(1) time. How can Takaoka’s method be applied to generating combinations and permutations? 4.14. The proof of Lemma 4.4 relies on Pascal’s formula, which states that for any 4.5. Problems 179 positive integers n and r such that r ≤ n, the following identity holds: n+1 n n = + . r r−1 r Prove Pascal’s formula. 4.15. Let m, n, r be nonnegative integers such that r ≤ n. Prove the Vandermonde convolution X r m+n m n = . r k=0 k r−k The latter equation, also known as Vandermonde’s identity, was already known as early as 1303 in China by Chu Shi-Chieh. Alexandre-Théophile Vandermonde independently discovered it and his result was published in 1772. 4.16. If m and n are nonnegative integers, prove that X n m+n+1 m+k = . n k=0 k 4.17. Let n be a positive integer. How many distinct binomial heaps having n vertices are there? 4.18. The algorithms described in section 4.3 are formally for minimum binomial heaps because the vertex at the top of the heap is always the minimum vertex. Describe analogous algorithms for maximum binomial heaps. 4.19. If H is a binomial heap, what is the total time required to extract all elements from H? 4.20. Frederickson [81] describes an O(k) time algorithm for finding the k-th smallest element in a binary heap. Provide a description and pseudocode of Frederickson’s algorithm and prove its correctness. 4.21. Fibonacci heaps [82] allow for amortized O(1) time with respect to finding the minimum element, inserting an element, and merging two Fibonacci heaps. Delet- ing the minimum element takes amortized time O(lg n), where n is the number of vertices in the heap. Describe and provide pseudocode of the above Fibonacci heap operations and prove the correctness of the procedures. 4.22. Takaoka [180] introduces another type of heap called a 2-3 heap. Deleting the minimum element takes amortized O(lg n) time with n being the number of vertices in the 2-3 heap. Inserting an element into the heap takes amortized O(1) time. Describe and provide pseudocode of the above 2-3 heap operations. Under which conditions would 2-3 heaps be more efficient than Fibonacci heaps? 4.23. In 2000, Chazelle [50] introduced the soft heap, which can perform common heap operations in amortized O(1) time. He then applied [49] the soft heap to realize a very efficient implementation of an algorithm for finding minimum spanning trees. In 2009, Kaplan and Zwick [117] provided a simple implementation and analysis of Chazelle’s soft heap. Describe soft heaps and provide pseudocode of common heap 180 Chapter 4. Graph data structures operations. Prove the correctness of the algorithms and provide runtime analyses. Describe how to use soft heap to realize an efficient implementation of an algorithm to produce minimum spanning trees. 4.24. Explain any differences between the binary heap-order property, the binomial heap- order property, and the binary search tree property. Can in-order traversal be used to list the vertices of a binary heap in sorted order? Explain why or why not. 4.25. Present pseudocode of an algorithm to find a vertex with maximum key in a binary search tree. 4.26. Compare and contrast algorithms for locating minimum and maximum elements in a list with their counterparts for a binary search tree. 4.27. Let T be a nonempty BST and suppose v ∈ V (T ) is not a minimum vertex of T . If h is the height of T , describe and present pseudocode of an algorithm to find the predecessor of v in worst-case time O(h). 4.28. Let L = [v0 , v1 , . . . , vn ] be the in-order listing of a BST T . Present an algorithm to find the successor of v ∈ V (T ) in constant time O(1). How can we find the predecessor of v in constant time as well? 4.29. Modify Algorithm 4.15 to extract a minimum vertex of a binary search tree. Now do the same to extract a maximum vertex. How can Algorithm 4.15 be modified to extract a vertex from a binary search tree? 4.30. Let v be a vertex of a BST and suppose v has two children. If s and p are the successor and predecessor of v, respectively, show that s has no left-child and p has no right-child. 4.31. Let L = [e0 , e1 , . . . , en ] be a list of n + 1 elements from a totally ordered set X with total order ≤. How can binary search trees be used to sort L? 4.32. Describe and present pseudocode of a recursive algorithm for each of the following operations on a BST. (a) Find a vertex with a given key. (b) Locate a minimum vertex. (c) Locate a maximum vertex. (d) Insert a vertex. 4.33. Are the algorithms presented in section 4.4 able to handle a BST having duplicate keys? If not, modify the relevant algorithm(s) to account for the case where two vertices in a BST have the same key. 4.34. The notion of vertex level for binary trees can be extended to general rooted trees as follows. Let T be a rooted tree with n > 0 vertices and height h. Then level 0 ≤ i ≤ h of T consists of all those vertices in T that have the same depth i. If each vertex at level i has i + m children for some fixed integer m > 0, what is the number of vertices at each level of T ? 4.5. Problems 181 4.35. Compare the search, insertion, and deletion times of AVL trees and random binary search trees. Provide empirical results of your comparative study. 4.36. Describe and present pseudocode of an algorithm to construct a Fibonacci tree of height n for some integer n ≥ 0. Analyze the worst-case runtime of your algorithm. 4.37. The upper bound in Theorem ?? can be improved as follows. From the proof of the theorem, we have the recurrence relation N (h) > N (h − 1) + N (h − 2). (a) If h ≤ 2, show that there exists some c > 0 such that N (h) ≥ ch . (b) Assume for induction that N (h) > N (h − 1) + N (h − 2) ≥ ch−1 + ch−2 for some h > 2. If c > 0, show that c2 − c − 1 = 0 is a solution to the recurrence relation ch−1 + ch−2 and that √ !h 1+ 5 N (h) > . 2 (c) Use the previous two parts to show that 1 h< · lg n lg ϕ √ where ϕ = (1 + 5)/2 is the golden ratio and n counts the number of internal vertices of an AVL tree of height h. 4.38. The Fibonacci sequence Fn is defined as follows. We have initial values F0 = 0 and F1 = 1. For n > 1, the n-th term in the sequence can be obtained via the recurrence relation Fn = Fn−1 + Fn−2 . Show that ϕn − (−1/ϕ)n Fn = √ (4.3) 5 where ϕ is the golden ratio. The closed form solution (4.3) to the Fibonacci se- quence is known as Binet’s formula, named after Jacques Philippe Marie Binet, even though Abraham de Moivre knew about this formula long before Binet did. Chapter 5 Distance and connectivity — Spiked Math, http://spikedmath.com/382.html 5.1 Paths and distance 5.1.1 Distance and metrics Consider an edge-weighted simple graph G = (V, E, i, h) without negative weight cycles. Here E ⊆ V (2) , i : E → V (2) is an incidence function as in (1.2), which we regard as the identity function, and h : E → V is an orientation function as in (1.3). Let W : E → R be the weight function. (If G is not provided with a weight function on the edges, we assume that each edge has unit weight.) If v1 , v2 ∈ V are two vertices and P = (e1 , e2 , . . . , em ) is a v1 -v2 path (so v1 is incident to e1 and v2 is incident to em ), define the weight of P to be the sum of the weights of the edges in P : m X W (P ) = W (ei ). i=1 The distance function d : V × V → R ∪ {∞} on G is defined by d(v1 , v2 ) = ∞ if v1 and v2 lie in distinct connected components of G, and by d(v1 , v2 ) = min W (P ) (5.1) P 182 5.1. Paths and distance 183 otherwise, where the minimum is taken over all paths P from v1 to v2 . By hypothesis, G has no negative weight cycles so the minimum in (5.1) exists. It follows by definition of the distance function that d(u, v) = ∞ if and only if there is no path between u and v. How we interpret the distance function d depends on the meaning of the weight function W . In practical applications, vertices can represent physical locations such as cities, sea ports, or landmarks. An edge weight could be interpreted as the physical distance in kilometers between two cities, the monetary cost of shipping goods from one sea port to another, or the time required to travel from one landmark to another. Then d(u, v) could mean the shortest route in kilometers between two cities, the lowest cost incurred in transporting goods from one sea port to another, or the least time required to travel from one landmark to another. The distance function d is not in general a metric, i.e. the triangle inequality does not in general hold for d. However, when the distance function is a metric then G is called a metric graph. The theory of metric graphs, due to their close connection with tropical curves, is an active area of research. For more information on metric graphs, see Baker and Faber [11]. 5.1.2 Radius and diameter A new hospital is to be built in a large city. Construction has not yet started and a number of urban planners are discussing the future location of the new hospital. What is a possible location for the new hospital and how are we to determine this location? This is an example of a class of problems known as facility location problems. Suppose our objective in selecting a location for the hospital is to minimize the maximum response time between the new hospital and the site of an emergency. To help with our decision making, we could use the notion of the center of a graph. The center of a graph G = (V, E) is defined in terms of the eccentricity of the graph under consideration. The eccentricity : V → R is defined as follows. For any vertex v, the eccentricity (v) is the greatest distance between v and any other vertex in G. In symbols, the eccentricity is expressible as (v) = max d(u, v). u∈V For example, in a tree T with root r the eccentricity of r is the height of T . In the graph of Figure 5.1, the eccentricity of 2 is 5 and the shortest paths that yield (2) are P1 : 2, 3, 4, 14, 15, 16 P2 : 2, 3, 4, 14, 15, 17. The eccentricity of a vertex v can be thought of as an upper bound on the distance from v to any other vertex in G. Furthermore, we have at least one vertex in G whose distance from v is (v). v 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 (v) 6 5 4 4 5 6 7 7 5 6 7 7 6 5 6 7 7 Table 5.1: Eccentricity distribution for the graph in Figure 5.1. To motivate the notion of the radius of a graph, consider the distribution of eccentric- ity among vertices of the graph G in Figure 5.1. The required eccentricity distribution 184 Chapter 5. Distance and connectivity 16 11 12 15 17 10 13 14 1 2 3 4 5 6 9 7 8 Figure 5.1: Determine the eccentricity, center, radius, and diameter. 7 6 (v) 5 4 0 5 10 15 v Figure 5.2: Eccentricity distribution of the graph in Figure 5.1. The horizontal axis represents the vertex name, while the vertical axis is the corresponding eccentricity. 5.1. Paths and distance 185 is shown in Table 5.1. Among the eccentricities in the latter table, the minimum eccen- tricity is (3) = (4) = 4. An intuitive interpretation is that both of the vertices 3 and 4 have the shortest distance to any other vertices in G. We can invoke an analogy with plane geometry as follows. If a circle has radius r, then the distance from the center of the circle to any point within the circle is at most r. The minimum eccentricity in graph theory plays a role similar to the radius of a circle. If an object is strategically positioned—e.g. a vertex with minimum eccentricity or the center of a circle—then its greatest distance to any other object is guaranteed to be minimum. With the above analogy in mind, we define the radius of a graph G = (V, E), written rad(G), to be the minimum eccentricity among the eccentricity distribution of G. In symbols, rad(G) = min (v). v∈V The center of G, written C(G), is the set of vertices with minimum eccentricity. Thus the graph in Figure 5.1 has radius 4 and center {3, 4}. As should be clear from the latter example, the radius is a number whereas the center is a set. Refer to the beginning of the section where we mentioned the problem of selecting a location for a new hospital. We could use a graph to represent the geography of the city wherein the hospital is to be situated and select a location that is in the center of the graph. Consider now the maximum eccentricity of a graph. In (3.5) we defined the diameter of a graph G = (V, E) by diam(G) = max d(u, v). u,v∈V u6=v The diameter of G can also be defined as the maximum eccentricity of any vertex in G: diam(G) = max (v). v∈V In case G is disconnected, define its diameter to be diam(G) = ∞. To compute diam(G), use the Floyd-Roy-Warshall algorithm (see section 3.6) to compute the shortest distance between each pair of vertices. The maximum of these distances is the diameter. The set of vertices of G with maximum eccentricity is called the periphery of G, written per(G). The graph in Figure 5.1 has diameter 7 and periphery {7, 8, 11, 12, 16, 17}. Theorem 5.1. Eccentricities of adjacent vertices. Let G = (V, E) be an undi- rected, connected graph having nonnegative edge weights. If uv ∈ E and W is a weight function for G, then |(u) − (v)| ≤ W (uv). Proof. By definition, we have d(u, x) ≤ (u) and d(v, x) ≤ (v) for all x ∈ V . Let w ∈ V such that d(u, w) = (u). Apply the triangle inequality to obtain d(u, w) ≤ d(u, v) + d(v, w) (u) ≤ W (uv) + d(v, w) ≤ W (uv) + (v) from which we have (u) − (v) ≤ W (uv). Repeating the above argument with the role of u and v interchanged yields (v) − (u) ≤ W (uv). Both (u) − (v) ≤ W (uv) and (v) − (u) ≤ W (uv) together yields the inequality |(u) − (v)| ≤ W (uv) as required. 186 Chapter 5. Distance and connectivity 5.1.3 Center of trees Given a tree T of order ≥ 3, we want to derive a bound on the number of vertices that comprise the center of T . A graph in general can have one, two, or more number of vertices for its center. Indeed, for any integer n > 0 we can construct a graph whose center has cardinality n. The cases for n = 1, 2, 3 are illustrated in Figure 5.3. But can we do the same for trees? That is, given any positive integer n does there exist a tree whose center has n vertices? It turns out that the center of a tree cannot have more than two vertices, a result first discovered [113] by Camille Jordan in 1869. (a) |C(G)| = 1 (b) |C(G)| = 2 (c) |C(G)| = 3 Figure 5.3: Constructing graphs with arbitrarily large centers. Theorem 5.2. Jordan [113]. If a tree T has order ≥ 3, then the center of T is either a single vertex or two adjacent vertices. Proof. As all eccentric vertices of T are leaves (see problem 5.7), removing all the leaves of T decreases the eccentricities of the remaining vertices by one. The tree comprised of the surviving vertices has the same center as T . Continue pruning leaves as described above and note that the tree comprised of the surviving vertices has the same center as the previous tree. After a finite number of leaf pruning stages, we eventually end up with a tree made up of either one vertex or two adjacent vertices. The vertex set of this final tree is the center of T . 5.1.4 Distance matrix In sections 1.3.4 and 3.3, the distance matrix D of a graph G was defined to be D = [dij ], where dij = d(vi , vj ) and the vertices of G are indexed by V = {v0 , v1 , . . . , vk }. The matrix D is square where we set dij = 0 for entries along the main diagonal. If there is no path from vi to vj , then we set dij = ∞. If G is undirected, then D is symmetric and is equal to its transpose, i.e. DT = D. To compute the distance matrix D, apply the Floyd-Roy-Warshall algorithm to determine the distances between all pairs of vertices. Refer to Figure 5.4 for examples of distance matrices of directed and undirected graphs. In the remainder of this section, “graph” refers to an undirected graph unless otherwise specified. Instead of one distance matrix, we can define several distance matrices on G. Consider an edge-weighted graph G = (V, E) without negative weight cycles and let d : V × V → R ∪ {∞} be a distance function of G. Let ∂ = diam(G) be the diameter of G and index the vertices of G in some arbitrary but fixed manner, say V = {v0 , v1 , . . . , vn }. The sequence 5.2. Vertex and edge connectivity 187 3 5 2 0 1 2 ∞ 1 2 ∞ 0 1 ∞ ∞ ∞ ∞ 1 0 ∞ ∞ ∞ ∞ 2 1 0 2 1 4 1 ∞ ∞ ∞ ∞ 0 1 0 ∞ ∞ ∞ ∞ 1 0 (a) 3 5 2 0 1 2 3 1 2 1 0 1 2 2 3 2 1 0 1 3 2 3 2 1 0 2 1 4 1 1 2 3 2 0 1 0 2 3 2 1 1 0 (b) Figure 5.4: Distance matrices of directed and undirected graphs. of distance matrices of G are a sequence of (n − 1) × (n − 1) matrices A1 , A2 , . . . , A∂ where ( 1, if d(vi , vj ) = k, (Ak )ij = 0, otherwise. In particular, A1 is the usual adjacency matrix A. To compute the sequence of distance matrices of G, use the Floyd-Roy-Warshall algorithm to compute the distance between each pair of vertices and assign the resulting distance to the corresponding matrix Ai . The distance matrix arises in several applications, including communication network design [89] and network flow algorithms [61]. Thanks to Graham and Pollak [89], the following unusual fact is known. If T is any tree then det D(T ) = (−1)n−1 (n − 1)2n−2 where n denotes the order of T . In particular, the determinant of the distance matrix of a tree is independent of the structure of the tree. This fact is proven in the paper [89], but see also [68]. 5.2 Vertex and edge connectivity If G = (V, E) is a graph and U ⊆ V is a vertex set with the property that G − U has more connected components than G, then we call U a vertex-cut. The term cut- vertex or cut-point is used when the vertex-cut consists of exactly one vertex. For an intuitive appreciation of vertex-cut, suppose G = (V, E) is a connected graph. Then U ⊆ V is a vertex-cut if the vertex deletion subgraph G − U is disconnected. For example, the cut-vertex of the graph in Figure 5.5 is the vertex 0. By κv (G) we mean the vertex connectivity of a connected graph G, defined as the minimum number of vertices whose removal would either disconnect G or reduce G to the trivial graph. The 188 Chapter 5. Distance and connectivity vertex connectivity κv (G) is also written as κ(G). The vertex connectivity of the graph in Figure 5.5 is κv (G) = 1 because we only need to remove vertex 0 in order to disconnect the graph. The vertex connectivity of a connected graph G is thus the vertex-cut of minimum cardinality. And G is said to be k-connected if κv (G) ≥ k. From the latter definition, it immediately follows that if G has at least 3 vertices and is k-connected then any vertex-cut of G has at least cardinality k. For instance, the graph in Figure 5.5 is 1-connected. In other words, G is k-connected if the graph remains connected even after removing any k − 1 or fewer vertices from G. 0 1 2 3 Figure 5.5: A claw graph with 4 vertices. Figure 5.6: The Petersen graph on 10 vertices. Example 5.3. Here is a Sage example concerning κ(G) using the Petersen graph de- picted in Figure 5.6. A linear programming Sage package, such as GLPK, must be installed for the commands below to work. sage : G = graphs . PetersenGraph () sage : len ( G . vertices ()) 10 sage : G . vertex_connectivity () 3.0 sage : G . delete_vertex (0) sage : len ( G . vertices ()) 9 sage : G . vertex_connectivity () 2.0 The notions of edge-cut and cut-edge are similarly defined. Let G = (V, E) be a graph and D ⊆ E an edge set such that the edge deletion subgraph G − D has more components than G. Then D is called an edge-cut. An edge-cut D is said to be minimal 5.2. Vertex and edge connectivity 189 if no proper subset of D is an edge-cut. The term cut-edge or bridge is reserved for the case where the set D is a singleton. Think of a cut-edge as an edge whose removal from a connected graph would result in that graph being disconnected. Going back to the case of the graph in Figure 5.5, each edge of the graph is a cut-edge. A graph having no cut-edge is called bridgeless. An open question as of 2010 involving bridges is the cycle double cover conjecture, due to Paul Seymour and G. Szekeres, which states that every bridgeless graph admits a set of cycles that contains each edge exactly twice. The edge connectivity of a connected graph G, written κe (G) and sometimes denoted by λ(G), is the minimum number of edges whose removal would disconnect G. In other words, κe (G) is the minimum cardinality among all edge-cuts of G. Furthermore, G is said to be k-edge-connected if κe (G) ≥ k. A connected graph that is k-edge-connected is guaranteed to be connected after removing ≤ k − 1 edges from it. When we have removed k or more edges, then the graph would become disconnected. By convention, a 1-edge-connected graph is simply a connected graph. The graph in Figure 5.5 has edge connectivity κe (G) = 1 and is 1-edge-connected. Example 5.4. Here is a Sage example concerning λ(G) using the Petersen graph shown in Figure 5.6. You must install an optional linear programming Sage package such as GLPK for the commands below to work. sage : G = graphs . PetersenGraph () sage : len ( G . vertices ()) 10 sage : E = G . edges (); len ( E ) 15 sage : G . edge_connectivity () 3.0 sage : G . delete_edge ( E [0]) sage : len ( G . edges ()) 14 sage : G . edge_connectivity () 2.0 Vertex and edge connectivity are intimately related to the reliability and survivability of computer networks. If a computer network G (which is a connected graph) is k- connected, then it would remain connected despite the failure of at most k − 1 network nodes. Similarly, G is k-edge-connected if the network remains connected after the failure of at most k − 1 network links. In practical terms, a network with redundant nodes and/or links can afford to endure the failure of a number of nodes and/or links and still be connected, whereas a network with very few redundant nodes and/or links (e.g. something close to a spanning tree) is more prone to be disconnected. A k-connected or k-edge-connected network is more robust (i.e. can withstand) against node and/or link failures than is a j-connected or j-edge-connected network, where j < k. Proposition 5.5. If δ(G) is the minimum degree of an undirected connected graph G = (V, E), then the edge connectivity of G satisfies λ(G) ≤ δ(G). Proof. Choose a vertex v ∈ V whose degree is deg(v) = δ(G). Deleting the δ(G) edges incident on v suffices to disconnect G as v is now an isolated vertex. It is possible that G has an edge-cut whose cardinality is smaller than δ(G). Hence the result follows. Let G = (V, E) be a graph and suppose X1 and X2 comprise a partition of V . A partition-cut of G, denoted hX1 , X2 i, is the set of all edges of G with one endpoint in X1 and the other endpoint in X2 . If G is a bipartite graph with bipartition X1 and X2 , then hX1 , X2 i is a partition-cut of G. It follows that a partition-cut is also an edge-cut. 190 Chapter 5. Distance and connectivity Proposition 5.6. An undirected connected graph G is k-edge-connected if and only if any partition-cut of G has at least k edges. Proof. Assume that G is k-edge-connected. Then each edge-cut has at least k edges. As a partition-cut is an edge-cut, then any partition-cut of G has at least k edges. On the other hand, suppose each partition-cut has at least k edges. If D is a minimal edge-cut of G and X1 and X2 are the vertex sets of the two components of G − D, then D = hX1 , X2 i. To see this, note that D ⊆ hX1 , X2 i. If hX1 , X2 i − D 6= ∅ then choose some e ∈ hX1 , X2 i such that e ∈/ D. The endpoints of e belong to the same component of G − D, in contradiction of the definition of X1 and X2 . Thus any minimal edge-cut is a partition-cut and conclude that any edge-cut has at least k edges. Proposition 5.7. If G = (V, E) is an undirected connected graph with vertex connectivity κ(G) and edge connectivity λ(G), then we have κ(G) ≤ λ(G). Proof. Let S be an edge-cut of G with cardinality k = |S| = λ(G). Removing k suitably chosen vertices of G suffice to delete the edges of S and hence disconnect G. It is also possible to have a smaller vertex-cut elsewhere in G. Hence the inequality follows. Taking together Propositions 5.5 and 5.7, we have Whitney’s inequality. Theorem 5.8. Whitney’s inequality [195]. Let G be an undirected connected graph with vertex connectivity κ(G), edge connectivity λ(G), and minimum degree δ(G). Then we have the following inequality: κ(G) ≤ λ(G) ≤ δ(G). Proposition 5.9. Let G be an undirected connected graph that is k-connected for some k ≥ 3. If e is an edge of G, then the edge-deletion subgraph G − e is (k − 1)-connected. Proof. Let V = {v1 , v2 , . . . , vk−2 } be a set of k − 2 vertices in G − e. It suffice to show the existence of a u-v walk in (G − e) − V for any distinct vertices u and v in (G − e) − V . We need to consider two cases: (i) at least one of the endpoints of e is in V ; and (ii) neither endpoints of e is in V . (i) Assume that V has at least one endpoint of e. As G − V is 2-connected, any distinct pair of vertices u and v in G − V is connected by a u-v path that excludes e. Hence the u-v path is also in (G − e) − V . (ii) Assume that neither endpoints of e is in V . If u and v are distinct vertices in (G − e) − V , then either: (1) both u and v are endpoints of e; or (2) at least one of u and v is an endpoint of e. (1) Suppose u and v are both endpoints of e. As G is k-connected, then G has at least k + 1 vertices so that the vertex set of G − {v1 , v2 , . . . , vk−2 , u, v} is nonempty. Let w be a vertex of G − {v1 , v2 , . . . , vk−2 , u, v}. Then there is a u-w path in G − {v1 , v2 , . . . , vk−2 , v} and a w-v path in G − {v1 , v2 , . . . , vk−2 , u}. Neither the u-w nor the w-v paths contain e. The concatenation of these two paths is a u-v walk in (G − e) − V . (2) Now suppose at least one of u and v, say u, is an endpoint of e. Let w be the other endpoint of e. As G is k-connected, then G − {v1 , v2 , . . . , vk−2 , w} is connected and we can find a u-v path P in G − {v1 , v2 , . . . , vk−2 , w}. Furthermore P is a u-v path in G − {v1 , v2 , . . . , vk−2 } that neither contain w nor e. Hence P is a u-v path in (G − e) − V . 5.2. Vertex and edge connectivity 191 Conclude that G − e is (k − 1)-connected. Repeated application of Proposition 5.9 results in the following corollary. Corollary 5.10. Let G be an undirected connected graph that is k-connected for some k ≥ 3. If E is any set of m edges of G, for m ≤ k − 1, then the edge-deletion subgraph G − E is (k − m)-connected. What does it mean for a communications network to be fault-tolerant? In 1932, Hassler Whitney provided [195] a characterization of 2-connected graphs whereby he showed that a graph G is 2-connected if and only if each pair of distinct vertices in G has two different paths connecting those two vertices. A key to understanding Whitney’s characterization of 2-connected graphs is the notion of internal vertex of a path. Given a path P in a graph, a vertex along that path is said to be an internal vertex if it is neither the initial nor the final vertex of P . In other words, a path P has an internal vertex if and only if P has at least two edges. Building upon the notion of internal vertices, we now discuss what it means for two paths to be internally disjoint. Let u and v be distinct vertices in a graph G and suppose P1 and P2 are two paths from u to v. Then P1 and P2 are said to be internally disjoint if they do not share any common internal vertex. Two u-v paths are internally disjoint in the sense that both u and v are the only vertices to be found in common between those paths. The notion of internally disjoint paths can be easily extended to a collection of u-v paths. Whitney’s characterization essentially says that a graph is 2-connected if and only if any two u-v paths are internally disjoint. Consider the notion of internally disjoint paths within the context of communications network. As a first requirement for fault-tolerant communications network, we want the network to remain connected despite the failure of any network node. By Whitney’s characterization, this is possible if the original communications network is 2-connected. That is, we say that a communications network is fault-tolerant provided that any pair of distinct nodes is connected by two internally disjoint paths. The failure of any node should at least guarantee that any two distinct nodes are still connected. Theorem 5.11. Whitney’s characterization of 2-connected graphs [195]. Let G be an undirected connected graph having at least 3 vertices. Then G is 2-connected if and only if any two distinct vertices in G are connected by two internally disjoint paths. Proof. (⇐=) For the case of necessity, argue by contraposition. That is, suppose G is not 2-connected. Let v be a cut-vertex of G, from which it follows that G−v is disconnected. We can find two vertices w and x such that there is no w-x path in G − v. Therefore v is an internal vertex of any w-x path in G. (=⇒) For the case of sufficiency, let G be 2-connected and let u and v be any two distinct vertices in G. Argue by induction on d(u, v) that G has at least two internally disjoint u-v paths. For the base case, suppose u and v are connected by an edge e so that d(u, v) = 1. Adapt the proof of Proposition 5.9 to see that G − e is connected. Hence we can find a u-v path P in G − e such that P and e are two internally disjoint u-v paths in G. Assume for induction that G has two internally disjoint u-v paths where d(u, v) < k for some k ≥ 2. Let w and x be two distinct vertices in G such that d(w, x) = k and hence there is a w-x path in G of length k, i.e. we have a w-x path W : w = w1 , w2 , . . . , wk−1 , wk = x. 192 Chapter 5. Distance and connectivity Note that d(w, wk−1 ) < k and apply the induction hypothesis to see that we have two internally disjoint w-wk−1 paths in G; call these paths P and Q. As G is 2-connected, we have a w-x path R in G − wk−1 and hence R is also a w-x path in G. Let z be the vertex on R that immediately precedes x and assume without loss of generality that z is on P . We claim that G has two internally disjoint w-x paths. One of these paths is the concatenation of the subpath of P from w to z with the subpath of R from z to x. If x is not on Q, then construct a second w-x path, internally disjoint from the first one, as follows: concatenate the path Q with the edge wk−1 w. In case x is on Q, take the subpath of Q from w to x as the required second path. From Theorem 5.11, an undirected connected graph G is 2-connected if and only if any two distinct vertices of G are connected by two internally disjoint paths. In particular, let u and v be any two distinct vertices of G and let P and Q be two internally disjoint u-v paths as guaranteed by Theorem 5.11. Starting from u, travel along the path P to arrive at v. Then start from v and travel along the path Q to arrive at u. The concatenation of the internally disjoint paths P and Q is hence a cycle passing through u and v. We have proved the following corollary to Theorem 5.11. Corollary 5.12. Let G be an undirected connected graph having at least 3 vertices. Then G is 2-connected if and only if any two distinct vertices of G lie on a common cycle. The following theorem provides further characterizations of 2-connected graphs, in addition to Whitney’s characterization. Theorem 5.13. Characterizations of 2-connected graphs. Let G = (V, E) be an undirected connected graph having at least 3 vertices. Then the following are equivalent. 1. G is 2-connected. 2. If u, v ∈ V are distinct vertices of G, then u and v lie on a common cycle. 3. If v ∈ V and e ∈ E, then v and e lie on a common cycle. 4. If e1 , e2 ∈ E are distinct edges of G, then e1 and e2 lie on a common cycle. 5. If u, v ∈ V are distinct vertices and e ∈ E, then they lie on a common path. 6. If u, v, w ∈ V are distinct vertices, then they lie on a common path. 7. If u, v, w ∈ V are distinct vertices, then there is a path containing any two of these vertices but excluding the third. 5.3 Menger’s theorem Menger’s theorem has a number of different versions: an undirected, vertex-connectivity version; a directed, vertex-connectivity version; an undirected, edge-connectivity version; and a directed, edge-connectivity version. In this section, we will prove the undirected, vertex-connectivity version. But first, let’s consider a few technical results that will be of use for the purpose of this section. Let u and v be distinct vertices in a connected graph G = (V, E) and let S ⊆ V . Then S is said to be u-v separating if u and v lie in different components of the vertex 5.3. Menger’s theorem 193 deletion subgraph G − S. The vertices u and v are positioned such that after removing vertices in S from G and the corresponding edges, u and v are no longer connected nor strongly connected to each other. It is clear by definition that u, v ∈ / S. We also say that S separates u and v, or S is a vertex separating set. Similarly an edge set T ⊆ E is u-v separating (or separates u and v) if u and v lie in different components of the edge deletion subgraph G − T . But unlike the case of vertex separating sets, it is possible for u and v to be endpoints of edges in T because the removal of edges does not result in deleting the corresponding endpoints. The set T is also called an edge separating set. In other words, S is a vertex cut and T is an edge cut. When it is clear from context, we simply refer to a separating set. See Figure 5.7 for illustrations of separating sets. (a) Original graph. (b) Vertex separated. (c) Original graph. (d) Edge separated. Figure 5.7: Vertex and edge separating sets. Blue-colored vertices are those we want to separate. The red-colored vertices form a vertex separating set or vertex cut; the red-colored edges constitute an edge separating set or edge cut. Proposition 5.14. Consider two distinct, non-adjacent vertices u, v in a connected graph G. If Puv is a collection of internally disjoint u-v paths in G and Suv is a u- v separating set of vertices in G, then |Puv | ≤ |Suv |. (5.2) Proof. Each u-v path in Puv must include at least one vertex from Suv because Suv is a vertex cut of G. Any two distinct paths in Puv cannot contain the same vertex from Suv . Thus the number of internally disjoint u-v paths is at most |Suv |. The bound (5.2) holds for any u-v separating set Suv of vertices in G. In particular, we can choose Suv to be of minimum cardinality among all u-v separating sets of vertices in G. Thus we have the following corollary. Menger’s Theorem 5.18 provides a much stronger statement of Corollary 5.15, saying in effect that the two quantities max(|Puv |) and min(|Suv |) are equal. Corollary 5.15. Consider any two distinct, non-adjacent vertices u, v in a connected graph G. Let max(|Puv |) be the maximum number of internally disjoint u-v paths in G and denote by min(|Suv |) the minimum cardinality of a u-v separating set of vertices in G. Then we have max(|Puv |) ≤ min(|Suv |). 194 Chapter 5. Distance and connectivity Corollary 5.16. Consider any two distinct, non-adjacent vertices u, v in a connected graph G. Let Puv be a collection of internally disjoint u-v paths in G and let Suv be a u-v separating set of vertices in G. If |Puv | = |Suv |, then Puv has maximum cardinality among all collections of internally disjoint u-v paths in G and Suv has minimum cardinality among all u-v separating sets of vertices in G. Proof. Argue by contradiction. Let Quv be another collection of internally disjoint u-v paths in G such that |Quv | ≥ |Puv |. Then |Puv | ≤ |Quv | ≤ |Suv | by Proposition 5.14. We cannot have |Quv | > |Puv |, which would be contradictory to our hypothesis that Puv = |Suv |. Thus |Quv | = |Puv |. Let Tuv be another u-v separating set of vertices in G such that |Tuv | ≤ |Suv |. Then we have |Puv | ≤ |Tuv | ≤ |Suv | by Proposition 5.14. We cannot have |Tuv | < |Suv | because we would then end up with |Puv | ≤ |Tuv | and Puv = |Suv |, a contradiction. Therefore |Tuv | = |Suv |. Lemma 5.17. Consider two distinct, non-adjacent vertices u, v in a connected graph G and let k be the minimum number of vertices required to separate u and v. If G has a u-v path of length 2, then G has k internally disjoint u-v paths. Proof. Argue by induction on k. For the base case, assume k = 1. Hence G has a cut vertex x such that u and v are disconnected in G − x. Any u-v path must contain x. In particular, there can be only one internally disjoint u-v path. Assume for induction that k ≥ 2. Let P : u, x, v be a path in G having length 2 and suppose S is a smallest u-v separating set for G − x. Then S ∪ {x} is a u-v separating set for G. By the minimality of k, we have |S| ≥ k − 1. By the induction hypothesis, we have at least k − 1 internally disjoint u-v paths in G − x. As P is internally disjoint from any of the latter paths, conclude that G has k internally disjoint u-v paths. Theorem 5.18. Menger’s theorem. Let G be an undirected connected graph and let u and v be distinct, non-adjacent vertices of G. Then the maximum number of internally disjoint u-v paths in G equals the minimum number of vertices needed to separate u and v. Proof. Suppose that the maximum number of independent u-v paths in G is attained by u-v paths P1 , . . . , Pk . To obtain a separating set W ⊂ V , we must at least remove one point in each path Pi . This implies the minimum number of vertices needed to separate u and v is at least k. Therefore, we have an upper bound: #{indep. u − v paths} ≤ #{min. number of vertices needed to separate u and v}. We show that equality holds. Let n denote the number of edges of G. The proof is by induction on n. By hypothesis, n ≥ 2. If n = 2 the statement holds by inspection, since in that case G is a line graph with 3 vertices V = {u, v, w} and 2 edges, E = {uw.wv}. In that situation, there is only 1 u-v path (namely, uwv) and only one vertex separating u and v (namely, w). Suppose now n > 3 and assume the statement holds for each graph with < n edges. Let k = #{independent u − v paths} and let ` = #{min. number of vertices needed to separate u and v}, 5.4. Whitney’s Theorem 195 so that k ≤ `. Let e ∈ E and let G/e be the contraction graph having edges E − {e} and vertices the same as those of G, except that the endpoints of e have been identified. Suppose that k < ` and G does not have ` independent u-v paths. The contraction graph G/e does not have ` independent u-v paths either (where now, if e contains u or v then we must appropriately redefine u or v, if needed). However, by the induction hypothesis G/e does have the property that the maximum number of internally disjoint u-v paths equals the minimum number of vertices needed to separate u and v. Therefore, #{independent u − v paths in G/e} < #{min. number of vertices needed to separate u and v in G}. By induction, #{independent u − v paths in G/e} = #{min. number of vertices needed to separate u and v in G/e}. Now, we claim we can pick e such that e does contain u or v and in such a way that #{minimum number of vertices needed to separate u and v in G} ≥ #{minimum number of vertices needed to separate u and v in G/e}. Proof: Indeed, since n > 3 any separating set realizing the minimum number of vertices needed to separate u and v in G cannot contain both a vertex in G adjacent to u and a vertex in G adjacent to v. Therefore, we may pick e accordingly. (Q.E.D. claim) The result follows from the claim and the above inequalities. The following statement is the undirected, edge-connectivity version of Menger’s the- orem. Theorem 5.19. Menger’s theorem (edge-connectivity form). Let G be an undi- rected graph, and let s and t be vertices in G. Then, the maximum number of edge- disjoint (s, t)-paths in G equals the minimum number of edges from E(G) whose deletion separates s and t. This is proven the same way as the previous version but using the generalized min- cut/max-flow theorem (see Remark 9.16 above). Theorem 5.20. Dirac’s theorem. Let G = (V, E) be an undirected k-connected graph with |V | ≥ k + 1 vertices for k ≥ 3. If S ⊆ V is any set of k vertices, then G has a cycle containing the vertices of S. Proof. 5.4 Whitney’s Theorem Theorem 5.21. Whitney’s theorem (vertex version). Suppose G = (V, E) is a graph with |V | ≥ k + 1. The following are equivalent: G is k-vertex-connected, Any pair of distinct vertices v, w ∈ V are connected by at least k independent paths. 196 Chapter 5. Distance and connectivity Solution. ... Theorem 5.22. Whitney’s theorem (edge version). Suppose G = (V, E) is a graph with |V | ≥ k + 1. The following are equivalent: the graph G is k-edge-connected, any pair of vertices are connected by at least k edge-disjoint paths. Solution. ... Theorem 5.23. Whitney’s Theorem. Let G = (V, E) be a connected graph such that |V | ≥ 3. Then G is 2-connected if and only if any pair u, v ∈ V has two internally disjoint paths between them. 5.5 Centrality of a vertex Louis, I think this is the beginning of a beautiful friendship. — Rick from the 1942 film Casablanca degree centrality betweenness centrality; for efficient algorithms, see [36, 200] closeness centrality eigenvector centrality The degree centrality of a graph G = (V, E) is the list parameterized by the vertex set V of G whose v-th entry is the fraction of vertices connected to v ∈ V . The centrality of a vertex within a graph determines the relative importance of that vertex to its graph. Degree centrality measures the number of edges incident upon a vertex. sage : G = graphs . R a n d o m N e w m an W a t t s S t r o g a t z (6 ,2 ,1/2) sage : G Graph on 6 vertices sage : D = G . degree_sequence () sage : D [5 , 4 , 3 , 3 , 3 , 2] sage : VG = G . vertices () sage : VG [0 , 1 , 2 , 3 , 4 , 5] sage : DC = [ QQ ( x )/ len ( VG ) for x in D ] sage : DC [5/6 , 2/3 , 1/2 , 1/2 , 1/2 , 1/3] This graph is shown in Figure 5.8. The closeness centrality is defined to be 1 . average distance to all vertices Closeness centrality is an inverse measure of centrality in that a larger value indicates a less central vertex while a smaller value indicates a more central vertex. 5.5. Centrality of a vertex 197 Figure 5.8: A randomly choosen graph whose “central-most” vertex is 0, with degree centrality equal to 5/6. Algorithm 5.1 Friendship graph. Input A positive integer n. Output The friendship graph Fn . 1: if n = 1 then 2: return C3 3: G ← null graph 4: N ← 2n + 1 5: for i ← 0, 1, . . . , N − 3 do 6: if i is odd then 7: add edges (i, i + 1) and (i, N − 1) to G 8: else 9: add edge (i, N − 1) to G 10: add edges (N − 2, 0) and (N − 2, N − 1) to G 11: return E 198 Chapter 5. Distance and connectivity 5.6 Network reliability Whitney synthesis Tutte’s synthesis of 3-connected graphs Harary graphs constructing an optimal k-connected n-vertex graph 5.7 The spectrum of a graph We use the notes “The spectrum of a graph” by Andries Brouwer [40] as a basic reference. Spectrum of a graph Laplacian spectrum of a graph Applications Examples from Boolean functions Let G = (V, E) be a (possibly directed) finite graph on n = |V | vertices. The adjacency matrix of G is the n × n matrix A = A(G) = (av,w )v,w∈V with rows and columns indexed by V and entries av,w denoting the number of edges from v to w. The spectrum of G, spec(G), is by definition the spectrum of A, that is, its multi-set of eigenvalues together with their multiplicities. The characteristic polynomial of G is that of A, that is, the polynomial pA defined by pA (x) = det(A − xI). 5.7.1 The Laplacian spectrum Recall from section 1.3.3 that, given a simple graph G with n vertices V = {v1 , . . . , vn }, its (vertex) Laplacian matrix L = (`i,j )n×n is defined as: deg(vi ), if i = j, `i,j = −1, if i 6= j and vi is adjacent to vj , 0, otherwise. The Laplacian spectrum is by definition the spectrum of the vertex Laplacian of G, that is, its multi-set of eigenvalues together with their multiplicities. For a graph G and its Laplacian matrix L with eigenvalues λn ≤ λn−1 ≤ · · · ≤ λ1 : For all i, λi ≥ 0 and λn = 0. The number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph. λn = 0 because every Laplacian matrix has an eigenvector of [1, 1, . . . , 1], 5.7. The spectrum of a graph 199 If we define a signed edge adjacency matrix M with element me,v for edge e ∈ E (connecting vertex vi and vj , with i < j) and vertex v ∈ V given by 1, if v = vi , Mev = −1, if v = vj , 0, otherwise then the Laplacian matrix L satisifies L = M T M , where M T is the matrix trans- pose of M . These are left as an exercise. 5.7.2 Applications of the (ordinary) spectrum The following is a basic fact about the largest eigenvalue of a graph. Theorem 5.24. Each graph G has a real eigenvalue λ1 > 0 with nonnegative real cor- responding eigenvector, and such that for each eigenvalue λ we have |λ| ≤ λ1 . We shall mostly be interested in the case where G is undirected, without loops or multiple edges. This means that A is symmetric, has zero diagonal (av,v = 0), and is a 0-1 matrix ( av,w ∈ {0, 1}). A number λ is eigenvalue of A if and only if it is a zero of the polynomial pA . Since A is real and symmetric, all its eigenvalues are real and A is diagonalizable. In particular, for each eigenvalue, its algebraic multiplicity (that is, its multiplicity as a root of the characteristic polynomial) coincides with its geometric multiplicity (that is, the dimension of the corresponding eigenspace). Theorem 5.25. Let G be a connected graph of diameter d. Then G has at least d + 1 distinct eigenvalues. Proof. Let the distinct eigenvalues of the adjacency matrix A of G be λ1 , ..., λr . Then (A − λ1 I)...(A − λr I) = 0, so that Ar is a linear combination of I, A, ..., Ar−1 . But if the distance from the vertex v ∈ V to the vertex w ∈ V is r, then (Ai )v,w = 0 for 0 ≤ i ≤ r − 1 and (Ar )v,w > 0, contradiction. Hence d > r. 5.7.3 Examples from Boolean functions Let f be a Boolean function on GF (2)n . The Cayley graph of f is defined to be the graph Γf = (GF (2)n , Ef ), whose vertex set is GF (2)n and the set of edges is defined by Ef = {(u, v) ∈ GF (2)n × GF (2)n | f (u + v) = 1}. The adjacency matrix Af of this graph is the matrix whose entries are Ai,j = f (b(i) + b(j)), 200 Chapter 5. Distance and connectivity where b(k) is the binary representation of the integer k. Note Γf is a regular graph of degree wt(f ), where wt denotes the Hamming weight of f when regarded as a vector of values (of length 2n ). Recall that, given a graph Γ and its adjacency matrix A, the spectrum Spec(Γ) is the multi-set of eigenvalues of A. The Walsh transform of a Boolean function f is an integer-valued function over GF (2)n that can be defined as X Wf (u) = (−1)f (x)+hu,xi . xinGF (2)n A Boolean function f is bent if |Wf (a)| = 2n/2 (this only makes sense if n is even). This property says, roughly speaking, that f is “as non-linear as possible.” The Hadamard transform of a integer-valued function f is an integer-valued function over GF (2)n that can be defined as X Hf (u) = f (x)(−1)hu,xi . xinGF (2)n It turns out that the spectrum of Γf is equal to the Hadamard transform of f when regarded as a vector of (integer) 0, 1-values. (This nice fact seems to have first appeared in [23], [24].) Recall that a graph is regular of degree r (or r-regular) if every vertex has degree r. We say that an r-regular graph Γ is a strongly regular graph with parameters (v, r, d, e) (for nonnegative integers e, d) provided, for all vertices u, v the number of vertices adjacent to both u, v is equal to ( e, if u, v are adjacent, d, if u, v are nonadjacent. It turns out tht f is bent if and only if Γf is strongly regular and e = d (see [24], [7]). The following Sage computations illustrate these and other theorems in [177], [23], [24], [7]. First, consider the Boolean function f : GF (2)4 → GF (2) given by f (x0 , x1 , x2 ) = x0 x1 + x2 x3 . sage : V = GF (2)^4 sage : f = lambda x : x [0]* x [1]+ x [2]* x [3] sage : CartesianProduct ( range (16) , range (16)) Cartesian product of [0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15] , [0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15] sage : C = CartesianProduct ( range (16) , range (16)) sage : Vlist = V . list () sage : E = [( x [0] , x [1]) for x in C if f ( Vlist [ x [0]]+ Vlist [ x [1]])==1] sage : len ( E ) 96 sage : E = Set ([ Set ( s ) for s in E ]) sage : E = [ tuple ( s ) for s in E ] sage : Gamma = Graph ( E ) sage : Gamma Graph on 16 vertices sage : VG = Gamma . vertices () sage : L1 = [] sage : L2 = [] sage : for v1 in VG : ....: for v2 in VG : ....: N1 = Gamma . neighbors ( v1 ) ....: N2 = Gamma . neighbors ( v2 ) ....: if v1 in N2 : 5.7. The spectrum of a graph 201 ....: L1 = L1 +[ len ([ x for x in N1 if x in N2 ])] ....: if not ( v1 in N2 ) and v1 != v2 : ....: L2 = L2 +[ len ([ x for x in N1 if x in N2 ])] ....: ....: sage : L1 ; L2 [2 , 2 , 2 , 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] [2 , 2 , 2 , 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] This implies the graph is strongly regular with d = e = 2. Let us use Sage to determine some properties of this graph. sage : Gamma . spectrum () [6 , 2 , 2 , 2 , 2 , 2 , 2 , -2 , -2 , -2 , -2 , -2 , -2 , -2 , -2 , -2] sage : [ walsh_transform (f , a ) for a in V ] [4 , 4 , 4 , -4 , 4 , 4 , 4 , -4 , 4 , 4 , 4 , -4 , -4 , -4 , -4 , 4] sage : Omega_f = [ v for v in V if f ( v )==1] sage : len ( Omega_f ) 6 sage : Gamma . is_bipartite () False sage : Gamma . is_hamiltonian () True sage : Gamma . is_planar () False sage : Gamma . is_regular () True sage : Gamma . is_eulerian () True sage : Gamma . is_connected () True sage : Gamma . is_triangle_free () False sage : Gamma . diameter () 2 sage : Gamma . degree_sequence () [6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6] The picture of the graph is given in Figure 5.9. sage : H = matrix ( QQ , 16 , 16 , [( -1)^( Vlist [ x [0]]). dot_product ( Vlist [ x [1]]) for x in C ]) sage : H [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [ 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1] [ 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1] [ 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1] [ 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1] [ 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1] [ 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1] [ 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1] [ 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1] [ 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1] [ 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1] [ 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1] [ 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1] [ 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1] [ 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1] [ 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1] sage : flist = vector ( QQ , [ int ( f ( v )) for v in V ]) sage : H * flist (6 , -2 , -2 , 2 , -2 , -2 , -2 , 2 , -2 , -2 , -2 , 2 , 2 , 2 , 2 , -2) sage : A = matrix ( QQ , 16 , 16 , [ f ( Vlist [ x [0]]+ Vlist [ x [1]]) for x in C ]) sage : A . eigenvalues () [6 , 2 , 2 , 2 , 2 , 2 , 2 , -2 , -2 , -2 , -2 , -2 , -2 , -2 , -2 , -2] The Hadamard transform of the Boolean function does indeed determine the spec- 202 Chapter 5. Distance and connectivity Figure 5.9: A Cayley graph of a Boolean function. trum of its Cayley graph. Here is another example: f : GF (2)3 → GF (2) given by f (x0 , x1 , x2 ) = x0 x1 + x2 . sage : V = GF (2)^3 sage : f = lambda x : x [0]* x [1]+ x [2] sage : Omega_f = [ v for v in V if f ( v )==1] sage : len ( Omega_f ) 4 sage : C = CartesianProduct ( range (8) , range (8)) sage : Vlist = V . list () sage : E = [( x [0] , x [1]) for x in C if f ( Vlist [ x [0]]+ Vlist [ x [1]])==1] sage : E = Set ([ Set ( s ) for s in E ]) sage : E = [ tuple ( s ) for s in E ] sage : Gamma = Graph ( E ) sage : Gamma Graph on 8 vertices sage : sage : VG = Gamma . vertices () sage : L1 = [] sage : L2 = [] sage : for v1 in VG : ....: for v2 in VG : ....: N1 = Gamma . neighbors ( v1 ) ....: N2 = Gamma . neighbors ( v2 ) ....: if v1 in N2 : ....: L1 = L1 +[ len ([ x for x in N1 if x in N2 ])] ....: if not ( v1 in N2 ) and v1 != v2 : ....: L2 = L2 +[ len ([ x for x in N1 if x in N2 ])] ....: sage : L1 ; L2 [2 , 0 , 2 , 2 , 2 , 2 , 0 , 2 , 2 , 2 , 0 , 2 , 2 , 2 , 2 , 0 , 0 , 2 , 2 , 2 , 2 , 0 , 2 , 2 , 2 , 0 , 2 , 2 , 2 , 2 , 0 , 2] [2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2] This implies that the graph is not strongly regular, therefore f is not bent. (There are other reasons why f cannot be bent as well.) Again, let us use Sage to determine some properties of this graph. sage : Gamma . spectrum () [4 , 2 , 0 , 0 , 0 , -2 , -2 , -2] 5.8. Expander graphs and Ramanujan graphs 203 sage : sage : Gamma . is_bipartite () False sage : Gamma . is_hamiltonian () True sage : Gamma . is_planar () False sage : Gamma . is_regular () True sage : Gamma . is_eulerian () True sage : Gamma . is_connected () True sage : Gamma . is_triangle_free () False sage : Gamma . diameter () 2 sage : Gamma . degree_sequence () [4 , 4 , 4 , 4 , 4 , 4 , 4 , 4] sage : H = matrix ( QQ , 8 , 8 , [( -1)^( Vlist [ x [0]]). dot_product ( Vlist [ x [1]]) for x in C ]) sage : H [ 1 1 1 1 1 1 1 1] [ 1 -1 1 -1 1 -1 1 -1] [ 1 1 -1 -1 1 1 -1 -1] [ 1 -1 -1 1 1 -1 -1 1] [ 1 1 1 1 -1 -1 -1 -1] [ 1 -1 1 -1 -1 1 -1 1] [ 1 1 -1 -1 -1 -1 1 1] [ 1 -1 -1 1 -1 1 1 -1] sage : flist = vector ( QQ , [ int ( f ( v )) for v in V ]) sage : H * flist (4 , 0 , 0 , 0 , -2 , -2 , -2 , 2) sage : Gamma . spectrum () [4 , 2 , 0 , 0 , 0 , -2 , -2 , -2] sage : A = matrix ( QQ , 8 , 8 , [ f ( Vlist [ x [0]]+ Vlist [ x [1]]) for x in C ]) sage : A . eigenvalues () [4 , 2 , 0 , 0 , 0 , -2 , -2 , -2] Again, we see the Hadamard transform does indeed determine the graph spectrum. The picture of the graph is given in Figure 5.10. Figure 5.10: Another Cayley graph of a Boolean function. 5.8 Expander graphs and Ramanujan graphs In combinatorics, an expander graph is a sparse graph that has strong connectivity properties. Expander graphs have many applications - for example, to cryptography, 204 Chapter 5. Distance and connectivity and the theory of error-correcting codes. The edge expansion h(G) of a graph G = (V, E) is defined as |∂(S)| h(G) = min , |V | 0<|S|≤ 2 |S| where the minimum is over all nonempty sets S of at most |V |/2 vertices and ∂(S) is the edge boundary of S, i.e., the set of edges with exactly one endpoint in S. The vertex expansion (or vertex isoperimetric number) hout (G) of a graph G is defined as |∂out (S)| hout (G) = min , 0<|S|≤ |V | 2 |S| where ∂out (S) is the outer boundary of S, i.e., the set of vertices in V (G) \ S with at least one neighbor in S. sage : G = PSL (2 , 5) sage : X = G . cayley_graph () sage : V = X . vertices () sage : S = [ V [1] , V [3] , V [7] , V [10] , V [13] , V [ 14] , V [23]] sage : delS = X . edge_boundary ( S ) sage : edge_expan_XS = len ( delS )/ len ( S ); RR ( edge_expan_XS ) 1.00000000000000 sage : S = [ V [1] , V [3] , V [7] , V [12] , V [24] , V [37]] sage : delS = X . edge_boundary ( S ) sage : edge_expan_XS = len ( delS )/ len ( S ); RR ( edge_expan_XS ) 1.50000000000000 sage : S = [ V [2] , V [8] , V [13] , V [27] , V [32] , V [44] , V [57]] sage : delS = X . edge_boundary ( S ) sage : edge_expan_XS = len ( delS )/ len ( S ); RR ( edge_expan_XS ) 1.42857142857143 sage : S = [ V [0] , V [6] , V [11] , V [16] , V [21] , V [29] , V [35] , V [45] , V [53]] sage : delS = X . edge_boundary ( S ) sage : edge_expan_XS = len ( delS )/ len ( S ); RR ( edge_expan_XS ) 1.77777777777778 sage : n = len ( X . vertices ()) sage : J = range ( n ) sage : J30 = Subsets (J , int ( n /2)) sage : K = J30 . random_element () sage : K {0 , 2 , 3 , 4 , 5 , 6 , 8 , 9 , 11 , 13 , 16 , 18 , 19 , 21 , 24 , 25 , 26 , 28 , 29 , 30 , 36 , 37 , 38 , 40 , 42 , 45 , 46 , 49 , 53 , 57} sage : S = [ V [ i ] for i in K ] # 30 vertices , randomly selected sage : delS = [ v for v in V if min ([ X . distance (a , v ) for a in S ]) == 1] sage : RR ( len ( delS ))/ RR ( len ( S )) 0.8 00000000000000 A family G = {G1 , G2 , . . .} of d-regular graphs is an edge expander family if there is a constant c > 0 such that h(G) ≥ c for each G ∈ G. A vertex expander family is defined similarly, using hout (G) instead. 5.8.1 Ramanujan graphs Let G be a connected d-regular graph with n vertices, and let λ0 ≥ λ1 ≥ . . . ≥ λn−1 be the eigenvalues of the adjacency matrix of G. Because G is connected and d-regular, its eigenvalues satisfy d = λ0 > λ1 ≥ . . . ≥ λn−1 ≥ −d. Whenever there exists λi with |λi | < d, define λ(G) = max |λi |. |λi |<d √ A d-regular graph G is a Ramanujan graph if λ(G) is defined and λ(G) ≤ 2 d − 1. 5.9. Problems 205 Let q be a prime power such that q ≡ 1 (mod 4). Note that this implies that the finite field GF (q) contains a square root of −1. Now let V = GF (q) and E = {{a, b} ∈ GF (q) × GF (q) | (a − b) ∈ GF (q)× )2 }. This set is well defined since a − b = (−1) · (b − a), and since −1 is a square, it follows that a−b is a square if and only if b − a is a square. By definition G = (V, E) is the Paley graph of order q. The following facts are known about Paley graphs. √ q−1 −1± q The eigenvalues of Paley graphs are 2 (with multiplicity 1) and 2 (both with multiplicity q−1 2 ). It is known that a Paley graph is a Ramanujan graph. It is known that the family of Paley graphs of prime order is a vertex expander graph family. If q = pr , where p is prime, then Aut(G) has order q(q − 1)/2. Here is Sage code for the Paley graph1 : def Paley ( q ): K . <a > = GF ( q ) return Graph ([ K , lambda i , j : i != j and (i - j ). is_square ()]) Below is an example. sage : X = Paley (13) sage : X . vertices () [0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12] sage : X . i s_vertex_transitive () True sage : X . degree_sequence () [6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6 , 6] sage : X . spectrum () [6 , 1.302775637731995? , 1.302775637731995? , 1.302775637731995? , 1.302775637731995? , 1.302775637731995? , 1.302775637731995? , -2.302775637731995? , -2.302775637731995? , -2.302775637731995? , -2.302775637731995? , -2.302775637731995? , -2.302775637731995?] sage : G = X . automorphism_group () sage : G . cardinality () 78 sage : 13*12/2 78 5.9 Problems When you don’t share your problems, you resent hearing the problems of other people. — Chuck Palahniuk, Invisible Monsters, 1999 5.1. Let G = (V, E) be an undirected, unweighted simple graph. Show that V and the distance function on G form a metric space if and only if G is connected. 5.2. Let u and v be two distinct vertices in the same connected component of G. If P is a u-v path such that d(u, v) = (u), we say that P is an eccentricity path for u. (a) If r is the root of a tree, show that the end-vertex of an eccentricity path for r is a leaf. 1 Thanks to Chris Godsil; see [86]. 206 Chapter 5. Distance and connectivity (b) If v is a vertex of a tree distinct from the root r, show that any eccentricity path for v must contain r or provide an example to the contrary. (c) A vertex w is said to be an eccentric vertex of v if d(v, w) = (v). Intuitively, an eccentric vertex of v can be considered as being as far away from v as possible. If w is an eccentric vertex of v and vice versa, then v and w are said to be mutually eccentric. See Buckley and Lau [45] for detailed discussions of mutual eccentricity. If w is an eccentric vertex of v, explain why v is also an eccentric vertex of w or show that this does not in general hold. 5.3. If u and v are vertices of a connected graph G such that d(u, v) = diam(G), show that u and v are mutually eccentric. 5.4. If uv is an edge of a tree T and w is a vertex of T distinct from u and v, show that |d(u, w) − d(w, v)| = W (uv) with W (uv) being the weight of uv. 5.5. If u and v are vertices of a tree T such that d(u, v) = diam(T ), show that u and v are leaves. 5.6. Let v1 , v2 , . . . , vk be the leaves of a tree T . Show that per(T ) = {v1 , v2 , . . . , vk }. 5.7. Show that all the eccentric vertices of a tree are leaves. 5.8. If G is a connected graph, show that rad(G) ≤ diam(G) ≤ 2 · rad(G). 5.9. Let T be a tree of order ≥ 3. If the center of T has one vertex, show that diam(T ) = 2 · rad(T ). If the center of T has two vertices, show that diam(T ) = 2 · rad(T ) − 1. 5.10. Let G = (V, E) be a simple undirected, connected graph. Define the distance of a vertex v ∈ V by X d(v) = d(v, x) x∈V and define the distance of the graph G itself by 1X d(G) = d(v). 2 v∈V For any vertex v ∈ V , show that d(G) ≤ d(v) + d(G − v) with G − v being a vertex deletion subgraph of G. This result appeared in Entringer et al. [70, p.284]. 5.11. Determine the sequence of distance matrices for the graphs in Figure 5.4. 5.12. If G = (V, E) is an undirected connected graph and v ∈ V , prove the following vertex connectivity inequality: κ(G) − 1 ≤ κ(G − v) ≤ κ(G). 5.13. If G = (V, E) is an undirected connected graph and e ∈ E, prove the following edge connectivity inequality: λ(G) − 1 ≤ λ(G − e) ≤ λ(G). 5.9. Problems 207 17 5 3 39 4 38 42 49 40 36 11 26 18 51 12 37 8 7 47 30 2 24 48 43 33 41 45 27 19 14 32 15 52 9 44 13 20 23 10 6 46 25 29 28 34 31 53 0 16 35 1 50 21 22 Figure 5.11: Network of common grape cultivars. 208 Chapter 5. Distance and connectivity code name code name code name 0 Alicante Bouschet 1 Aramon 2 Bequignol 3 Cabernet Franc 4 Cabernet Sauvignon 5 Carignan 6 Chardonnay 7 Chenin Blanc 8 Colombard 9 Donzillinho 10 Ehrenfelser 11 Fer Servadou 12 Flora 13 Gamay 14 Gelber Ortlieber 15 Grüner Veltliner 16 Kemer 17 Merlot 18 Meslier-Saint-Francois 19 Müller-Thurgau 20 Muscat Blanc 21 Muscat Hamburg 22 Muscat of Alexandria 23 Optima 24 Ortega 25 Osteiner 26 Péagudo 27 Perle 28 Perle de Csaba 29 Perlriesling 30 Petit Manseng 31 Petite Bouschet 32 Pinot Noir 33 Reichensteiner 34 Riesling 35 Rotberger 36 Roter Veltliner 37 Rotgipfler 38 Royalty 39 Ruby Cabernet 40 Sauvignon Blanc 41 Schönburger 42 Semillon 43 Siegerrebe 44 Sylvaner 45 Taminga 46 Teinturier du Cher 47 Tinta Madeira 48 Traminer 49 Trincadeiro 50 Trollinger 51 Trousseau 52 Verdelho 53 Wittberger Table 5.2: Numeric code and actual name of common grape cultivars. 5.14. Figure 5.11 depicts how common grape cultivars are related to one another; the graph is adapted from Myles et al. [149]. The numeric code of each vertex can be interpreted according to Table 5.2. Compute various distance and connectivity measures for the graph in Figure 5.11. 5.15. Prove the characterizations of 2-connected graphs as stated in Theorem 5.13. 5.16. Let G = (V, E) be an undirected connected graph of order n and suppose that deg(v) ≥ (n + k − 2)/2 for all v ∈ V and some fixed positive integer k. Show that G is k-connected. 5.17. A vertex (or edge) separating set S of a connected graph G is minimum if S has the smallest cardinality among all vertex (respectively edge) separating sets in G. Similarly S is said to be maximum if it has the greatest cardinality among all vertex (respectively edge) separating sets in G. For the graph in Figure 5.7(a), determine the following: (a) A minimum vertex separating set. (b) A minimum edge separating set. (c) A maximum vertex separating set. (d) A maximum edge separating set. (e) The number of minimum vertex separating sets. (f) The number of minimum edge separating sets. Chapter 6 Centrality and prestige 6.1 Vertex centrality 6.2 Edge centrality 6.3 Ranking web pages 6.4 Hub and authority 6.5 Problems 6.1. A. 209 Chapter 7 Optimal graph traversals 7.1 Eulerian graphs Motivation: tracing out all the edges of a graph without lifting your pencil. multigraphs and simple graphs Eulerian tours Eulerian trails 7.2 Hamiltonian graphs — Randall Munroe, xkcd, http://xkcd.com/230/ Motivation: the eager tourist problem: visiting all major sites of a city in the least time/distance. Hamiltonian paths (or cycles) Hamiltonian graphs Theorem 7.1. Ore 1960. Let G be a simple graph with n ≥ 3 vertices. If deg(u) + deg(v) ≥ n for each pair of non-adjacent vertices u, v ∈ V (G), then G is Hamiltonian. Corollary 7.2. Dirac 1952. Let G be a simple graph with n ≥ 3 vertices. If deg(v) ≥ n/2 for all v ∈ V (G), then G is Hamiltonian. 210 7.3. The Chinese Postman Problem 211 7.3 The Chinese Postman Problem See section 6.2 of Gross and Yellen [91]. de Bruijn sequences de Bruijn digraphs constructing a (2, n)-de Bruijn sequence postman tours and optimal postman tours constructing an optimal postman tour 7.4 The Traveling Salesman Problem — Randall Munroe, xkcd, http://xkcd.com/399/ See section 6.4 of Gross and Yellen [91], and section 35.2 of Cormen et al. [57]. Gray codes and n-dimensional hypercubes the Traveling Salesman Problem (TSP) nearest neighbor heuristic for TSP some other heuristics for solving TSP Chapter 8 Graph coloring — Spiked Math, http://spikedmath.com/210.html See Jensen and Toft [108] for a survey of graph coloring problems. See Dyer and Frieze [66] for an algorithm on randomly colouring random graphs. Graph coloring problems originated with the coloring of maps. For example, regard each state in the United States as a vertex, and connect two vertices by an edge if and only if they share a boundary, i.e., are neighbors. If you can color the United States map using k colors in such a way that no two neighboring states have the same color then we say the map has a k-coloring. While a student in London in the mid-1800’s, the South African mathematician Francis Guthrie conjectured to his mathematics professor Augustus de Morgan that four colors suffice to color any map. It was an open problem for over 100 years (only proven by Appel and Haken in 1976). 8.1 Vertex coloring When used without any qualification, a coloring of an undirected graph G = (V, E) is intended to mean a vertex coloring, namely a labelling of the graph’s vertices with colors such that no two vertices sharing an edge have the same color. A coloring using at most k colors is called a (proper) k-coloring. For example, the 2-colorable graphs are exactly the bipartite graphs. 212 8.1. Vertex coloring 213 Remark 8.1. For each k = 3, 4, . . . , the corresponding decision problem of deciding if a given graph can be k-colored is NP-hard (see [106]). The smallest number of colors needed to color a graph G is called its (vertex) chro- matic number, and is here denoted χv (G). A subset of V assigned to the same color is called a color-class. A subset S of V is called an independent set if no two vertices in S are adjacent in G. By definition, every color-class forms an independent set. The set of t-colorings is in one-to-one correspondence with the partitions of V into t independent sets. Example 8.2. The Dyck graph, shown in Figure 8.1, is named after Walther von Dyck. It is a 3-regular graph with 32 vertices and 48 edges. The graph is Hamiltonian with 120 distinct Hamiltonian cycles. It has chromatic number 2 (in other words, is bipartite), chromatic index 3, radius 5, diameter 5 and girth 6. It is also a 3-vertex-connected and a 3-edge-connected graph. sage : G = graphs . LCFGraph (32 , [5 , -5 ,13 , -13] , 8) sage : G . is_bipartite () True sage : G . coloring () [[1 , 3 , 5 , 7 , 9 , 11 , 13 , 15 , 17 , 19 , 21 , 23 , 25 , 27 , 29 , 31] , [0 , 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 , 22 , 24 , 26 , 28 , 30]] sage : G . i s_vertex_transitive () True Figure 8.1: A Dyck graph example. A clique in G is a subset of the vertex set S ⊂ V , such that for every two vertices in S, there exists an edge connecting the two. The clique number ω(G) of a graph G is the number of vertices in a maximal clique (a clique which cannot be extended to a clique of larger size by adding a vertex to it) in G. From the definitions, we see that the chromatic number is at least the clique number: 214 Chapter 8. Graph coloring χv (G) ≥ ω(G). It is also not hard to give a non-trivial upper bound. Theorem 8.3. Every simple graph can be colored with one more color than the maximum vertex degree, χv (G) ≤ ∆(G) + 1. We give two proofs. Proof. (proof 1) We prove this by induction on the number n of vertices. It is obvious in the case n = 1. Assume the result is true for all graphs with k − 1 vertices and let G = (V, E) be a graph with k vertices. We want to chow that G has a coloring with 1 + ∆(G) colors. Let v ∈ V and consider the graph G − v. By hypothesis, this graph has a coloring with 1 + ∆(G − v) colors. Since there are at most ∆(G) neighbors of v, no more than ∆(G) colors could be used for these adjacent vertices. There are two cases. Case 1 + ∆(G − v) < 1 + ∆(G). In this case, we create a new color for v to obtain a coloring for G with 2 + ∆(G − v) ≤ 1 + ∆(G) colors. Case 1 + ∆(G − v) = 1 + ∆(G). In this case, the vertices adjacent to v have been colored with at most ∆(G) colors, leaving us with at least one unused color. We can use that color for v. Proof. (proof 2) The fact that the graph coloring (decision) problem is NP-complete must not prevent one from trying to color it greedily. One such method would be to iteratively pick, in a graph G, an uncolored vertex v, and to color it with the smallest color available which is not yet used by one of its neighbors. Such a coloring algorithm will never use more than ∆(G) + 1 different colors (where ∆(G) is the maximal degree of G), as no vertex in the procedure will ever exclude more than ∆(G) colors. Such a greedy algorithm can be written in Sage in a few lines: sage : g = graphs . RandomGNP (100 ,5/100) sage : C = Set ( xrange (100)) sage : color = {} sage : for u in g : ... interdits = Set ([ color [ v ] for v in g . neighbors ( u ) if color . has_key ( v )]) ... color [ u ] = min (C - interdits ) Example 8.4. A Frucht graph, shown in Figure 8.2, has 12 nodes and 18 edges. It is 3-regular, planar and Hamiltonian. It is named after Robert Frucht. The Frucht graph has no nontrivial symmetries. It has chromatic number 3, chromatic index 3, radius 3, diameter 4 and girth 3. It is 3-vertex-connected and 3-edge-connected graph. sage : G = graphs . FruchtGraph () sage : G . show ( dpi =300) sage : vc = G . coloring () sage : G . chromatic_number () 3 sage : d = { ’ blue ’: vc [0] , ’ red ’: vc [1] , ’ green ’: vc [2]} sage : G . show ( vertex_colors = d ) sage : G . a utomorphism_group (). order () 1 Brook’s Theorem 8.2. Edge coloring 215 Figure 8.2: Frucht graph vertex-coloring example. heuristics for vertex coloring Theorem 8.5. (Brooks’ inequality) If G is not a complete graph and is not an odd cycle graph then χv (G) ≤ ∆(G). 8.2 Edge coloring Edge coloring is the direct application of vertex coloring to the line graph of a graph G. (Recall, L(G) is the graph whose vertices are the edges of G, two vertices being adjacent if and only if their corresponding edges share an endpoint). We write χv (L(G)) = χe (G) for the chromatic index of G. (This is also called the edge chromatic number. An edge coloring of a graph is an assignment of colors to edges so that no vertex is incident to two edges of the same color. An edge coloring with k colors is called a k-edge-coloring. Example 8.6. The Heawood graph, shown in Figure 8.4, is named after Percy John Heawood. It is an undirected graph with 14 vertices and 21 edges. It is a 3-regular, distance-transitive, distance-regular graph. It has chromatic number 2 and chromatic index 3. An edge-coloring is shown in Figure 8.3. A vertex-coloring is shown in Figure 8.4. Recall that a graph is vertex-transitive if its automorphism group acts transitively upon its vertices. The automorphism group of the Heawood graph is isomorphic to the projective linear group PGL 2(7), a group of order 336. It acts transitively on the vertices and on the edges of the graph. Therefore, this graph is vertex-transitive. 216 Chapter 8. Graph coloring sage : G = graphs . HeawoodGraph () sage : ec = edge_coloring ( G ) sage : d = { ’ blue ’: ec [0] , ’ green ’: ec [1] , ’ red ’: ec [2]} sage : G . show ( edge_colors = d ) sage : G . line_graph (). chromatic_number () # chromatic index 3 Figure 8.3: The Heawood graph and an edge-coloring example. sage : G = graphs . HeawoodGraph () sage : vc = G . coloring () sage : vc [[1 , 3 , 5 , 7 , 9 , 11 , 13] , [0 , 2 , 4 , 6 , 8 , 10 , 12]] sage : d = { ’ blue ’: vc [0] , ’ red ’: vc [1]} sage : G . show ( vertex_colors = d ) sage : G . chromatic_number () 2 Example 8.7. The Icosahedral graph, shown in Figure 8.5, is a particular projection of the edges of the solid icosahedron onto the plane. It is a 4-regular graph with 30 edges and 12 vertices. It has chromatic number 4 and chromatic index 5. An edge-coloring is shown in Figure 8.6. A vertex-coloring is shown in Figure 8.5. sage : G = graphs . IcosahedralGraph () sage : G . chromatic_number () 4 sage : vc = G . coloring () sage : d = { ’ blue ’: vc [0] , ’ red ’: vc [1] , ’ green ’: vc [2] , ’ orange ’: vc [3]} sage : G . show ( vertex_colors = d ) sage : G . is_hamiltonian () True sage : G . is_regular () True sage : G . i s_vertex_transitive () True 8.2. Edge coloring 217 Figure 8.4: The Heawood graph and an vertex-coloring example. Figure 8.5: An icosahedral graph vertex-coloring example. 218 Chapter 8. Graph coloring sage : G . is_perfect () False sage : G . is_planar () True sage : G . is_clique () False sage : G . is_bipartite () False sage : G . line_graph (). chromatic_number () 5 sage : ec = edge_coloring ( G ) sage : d = { ’ blue ’: ec [0] , ’ red ’: ec [1] , ’ green ’: ec [2] , ’ orange ’: ec [3] , ’ yellow ’: ec [4]} sage : G . show ( edge_colors = d ) Figure 8.6: An icosahedral graph edge-coloring example. As in the case of vertex-coloring, the edge-coloring decision problem is still NP- complete. However, it is much better understood through Vizing’s theorem. Theorem 8.8. (Vizing’s theorem) The edges of a graph G can be properly colored using at least ∆(G) colors and at most ∆(G) + 1, ∆(G) ≤ χe (G) ≤ ∆(G) + 1. Notice that the lower bound can be easily proved : if a vertex v has a degree d(v), then at least d(v) colors are required to color G as all the edges incident to v must receive different colors. Note the upper bound of ∆(G) + 1 cannot be deduced from the greedy algorithm given in the previous section, as the maximal degree of the line graph L(G) is not equal to ∆(G) but to max d(u) + d(v) − 2, which can reach 2∆(G) − 2 in regular graphs. u∼v Example 8.9. The Pappus graph, shown in Figure 8.7, is named after Pappus of Alexan- dria. It is 3-regular, symmetric, and distance-regular with 18 vertices and 27 edges. The Pappus graph has girth 6, diameter 4, radius 4, chromatic number 2 (i.e. is bipartite), chromatic index 3 and is both 3-vertex-connected and 3-edge-connected. sage : G = graphs . PappusGraph () sage : G . coloring () [[1 , 3 , 5 , 6 , 8 , 10 , 12 , 14 , 16] , [0 , 2 , 4 , 7 , 9 , 11 , 13 , 15 , 17]] sage : G . is_regular () 8.3. The chromatic polynomial 219 True sage : G . is_planar () False sage : G . i s_vertex_transitive () True sage : G . is_hamiltonian () True sage : G . girth () 6 sage : G . is_bipartite () True sage : G . show ( dpi =300) sage : G . line_graph (). chromatic_number () 3 sage : ec = edge_coloring ( G ) sage : ec [[(0 , 1) , (2 , 3) , (4 , 5) , (6 , 17) , (7 , 14) , (8 , 13) , (9 , 16) , (10 , 15) , (11 , 12)] , [(0 , 5) , (1 , 2) , (3 , 4) , (6 , 13) , (7 , 12) , (8 , 15) , (9 , 14) , (10 , 17) , (11 , 16)] , [(0 , 6) , (1 , 7) , (2 , 8) , (3 , 9) , (4 , 10) , (5 , 11) , (12 , 15) , (13 , 16) , (14 , 17)]] sage : d = { ’ blue ’: ec [0] , ’ red ’: ec [1] , ’ green ’: ec [2]} sage : G . plot ( edge_colors = d ). show () Figure 8.7: A Pappas graph edge-coloring example. algorithm for edge coloring by maximum matching algorithm for sequential edge coloring 8.3 The chromatic polynomial George David Birkhoff introduced the chromatic polynomial in 1912. The chromatic polynomial is defined as the unique interpolating polynomial of degree n through the points (k, PG (k)) for k = 0, 1, . . . n, where n is the number of vertices in G. For natural 220 Chapter 8. Graph coloring numbers t, the chromatic polynomial is the function that counts the number of t-colorings of G. As the name indicates, for a given GQthe function is indeed a polynomial in t. For a complex number x, let x(k) = k−1 i=0 (x − i). Lemma 8.10. If mk (G) denotes the number of distinct partitions of V into k different color-classes then n X PG (t) = mk (G)t(k) . k=1 Proof. Given a partition of V into k color-classes, we can assign t colors to the color- classes in t(t − 1) . . . (t − k + 1) = t(k) ways. For each k with 1 ≤ k ≤ n, there are mk (G) distinct partitions Pn of V into k such color-classes. Therefore, the number of colorings with t colors is k=1 mk (G)t(k) , as desired. Theorem 8.11. A graph G with n vertices is a tree if and only if P (G, t) = t(t − 1)n−1 . Proof. We prove this by induction on n. The statement is clearly true when n = 1. Assume the statement holds for any tree of n − 1 vertices and let G be a tree with n vertices. There is a vertex v of G having degree 1. The graph G − v is a tree having n − 1 vertices, so has t(t − 1)n−2 t-colorings. The number of t-colorings of G is the number of t-colorings of G − v times the number of ways to color v. Since we may color v using any color not used for its one neighbor, there are t − 1 colorings of v. Therefore, G has t(t − 1)n−1 t-colorings. The theorem above gives examples of non-isomorphic graphs which have the same chromatic polynomial. Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. For example, two trees having the same number of vertices are chromatically equivalent. It is also an open problem to find necessary and sufficient conditions for two arbitrary graphs to be chromatically equivalent. Theorem 8.12. If G has n vertices, m edges, and k components G1 , G2 ,. . . ,Gk , then P (G, t) = P (G1 , t)P (G2 , t) · · · P (Gk , t). This is proven by induction on k and the proof is left to the interested reader. Fix a pair of vertices u, v ∈ V . Recall that the (edge contraction) graph G/uv is obtained by merging the two vertices and removing any edges between them. Recall that the (edge deletion) graph G − uv is obtained by merging the the edge uv but not the two vertices u, v. Theorem 8.13. (Fundamental Reduction Theorem) The chromatic polynomial satisfies the recurrence relation P (G, t) = P (G − uv, t) − P (G/uv, t). 8.4. Applications of graph coloring 221 For example, if G is a tree then G/uv is another tree but G − uv is a non-connected graph. A root (or zero) of a chromatic polynomial, called a chromatic root is a value x where PG (x) = 0. Theorem 8.14. χv (C) is the smallest positive integer that is not a chromatic root: χv (G) = min{k | P (G, k) > 0}. Birkhoff noted that one could establish the four color theorem by showing P (G, 4) > 0 for all planar graphs G. This lead to the following statement, which is still an open problem today. Conjecture 8.15. (Birkhoff-Lewis Conjecture) As a function of a real variable t, P (g, t) is zero-free in the interval t ≥ 4. 8.4 Applications of graph coloring assignment problems scheduling problems matching problems map coloring and the Four Color Problem 8.4.1 Application to scheduling Assume that we have a finite set of different people (for example, students) and a finite set of distinct one-hour meetings they could attend (for example, classes). Construct the graph G as follows. The vertices V of G are given by the set of meetings. We connect two meetings with an edge if some person needs to attend both meetings. Example 8.16. Suppose the × in the table below indicated that the student in that column is attending the class in that row. Bob Carol Ted Zoe Calculus × × Chemistry × × × × English × × × × History × × Physics × × Let 0 = Calculus, 1 = Chemistry, 2 = English, 3 = History, and 4 = Physics. Then the corresponding graph can be constructed and drawn in Sage using the following com- mands (see Figure 8.8). sage : G = Graph ({0:[1 ,2 ,4] , 1:[2 ,3 ,4] , 2:[3 ,4]}) sage : G . show () sage : G . coloring () [[2] , [4] , [0 , 3] , [1]] 222 Chapter 8. Graph coloring Figure 8.8: An application of graph coloring to scheduling. Theorem 8.17. The minimum number of hours required for the schedule of meetings in our scheduling problem is χv (G). Proof. Suppose we can schedule the meetings in k hours. In other words, each person attends the meetings they need to and they can do so in at most k kours. Order the meeting times 1, 2, . . . , k. Each meeting must occur in one and only one of these time slots (although a given time slot may have several meetings). We color the graph G as follows: if a meeting occurs in hour i then use color i for all the meetings that meet at that hour. Consider two adjacent vertices. These vertices correspond to two meetings which share one or more people. Since a person cannot be in two places at the same time, these two meetings by have different meeting times, hence the vertices must have different colors. This implies χv (G) ≤ k. Conversely, suppose that G has a k-coloring. The meetings with color i (where 1 ≤ i ≤ k) can be held at the same time since any two such meetings correspond to non-adjacnt vertices, hence have no person in common. Therefore, the minimum number of hours requires for the meeting schedule is less than or equal to χv (G). 8.1. What is the number of hours required for a schedule in Example 8.16? 8.2. Draw the Dyck graph in Example 8.2 as a bipartite graph. 8.3. Draw the Pappas graph in Example 8.9 as a bipartite graph. 8.4.2 Map coloring Theorem 8.18. (Four Color Theorem) Every planar graph can be 4-colored. Chapter 9 Network flows See Jungnickel [114], and chapter 12 of Gross and Yellen [91]. 9.1 Flows and cuts single source-single sink networks feasible networks maximum flow and minimum cut Let G = (V, E, i, h) be an unweighted multidigraph, as in Definition 1.6. If F is a field such as R or GF (q) or a ring such as Z, let C 0 (G, F ) = {f : V → F }, C 1 (G, F ) = {f : E → F }, be the sets of F -valued functions defined on V and E, respectively. If F is a field then these are F -inner product spaces with inner product X (f, g) = f (x)g(x), (X = V, resp. X = E), (9.1) x∈X and dim C 0 (G, F ) = |V |, dim C 1 (G, F ) = |E|. If you index the sets V and E in some arbitrary but fixed way and define, for 1 ≤ i ≤ |V | and 1 ≤ j ≤ |E|, 1, v = vi , 1, e = ej , fi (v) = gj (e) = 0, otherwise, 0, otherwise, then F = {fi } ⊂ C 0 (G, F ) is a basis of C 0 (G, F ) and G = {gj } ⊂ C 1 (G, F ) is a basis of C 1 (G, F ). We order the edges E = {e1 , e2 , . . . , e|E| }, in some arbitrary but fixed way. A vector representation (or characteristic vector or incidence vector) of a subgraph G0 = (V, E 0 ) of G = (V, E), E 0 ⊂ E, is a binary |E|-tuple 223 224 Chapter 9. Network flows vec(G0 ) = (a1 , a2 , . . . , a|E| ) ∈ GF (2)|E| , where 0 1, if ei ∈ E 0 , ai = ai (E ) = / E 0. 0, if ei ∈ In particular, this defines a mapping vec : {subgraphs of G = (V, E)} → GF (2)|E| . For any non-trivial partition V = V1 ∪ V2 , Vi 6= ∅, V1 ∩ V2 = ∅, the set of all edges e = (v1 , v2 ) ∈ E, with vi ∈ Vi (i = 1, 2), is called a cocycle1 of G. A cocycle with a minimal set of edges is a bond (or cut set) of G. An Euler subgraph is either a cycle or a union of edge-disjoint cycles. The set of cycles of G is denoted Z(G) and the set of cocycles is denoted Z ∗ (G). The F -vector space spanned by the vector representations of all the cycles is called the cycle space of G, denoted Z(G) = Z(G, F ). This is the kernel of the incidence matrix of G (§14.2 in Godsil and Royle [87]). Define 1 D : CP (G, F ) → C 0 (G, P F ), (Df )(v) = h(e)=v f (e) − t(e)=v f (e). With respect to these bases F and G, the matrix representing the linear transformation D : C 1 (G, F ) → C 0 (G, F ) is the incidence matrix. An element of the kernel of D is sometimes called a flow (see Biggs [25]) or circulation (see below). Therefore, this is sometimes also referred to as the space of flows or the circulation space. It may be regarded as a subspace of C 1 (G, F ) of dimension n(G). When F is a finite field, sometimes2 the cycle space is called the cycle code of G. Let F be a field such as R or GF (q), for some prime power q. Let G be a digraph. Some define a circulation (or flow) on G = (V, E) to be a function f : E → F, satisfying3 P P u∈V, (u,v)∈E f (u, v) = w∈V, (v,w)∈E f (v, w). (Note: this is simply the condition that f belongs to the kernel of D.) Suppose G has a subgraph H and f is a circulation of G such that f is a constant function on H and 0 elsewhere. We call such a circulation a characteristic function of H. For example, if G has a cycle C and if f is the characteristic function on C, then f is a circulation. The circulation space C is the F -vector space of circulation functions. The cycle space “clearly” may be identified with a subspace of the circulation space, since the F -vector 1 Also called an edge cut subgraph or disconnecting set or seg or edge cutset. 2 Jungnickel and Vanstone in [115] call this the even graphical code of G. 3 Note: In addition, some authors add the condition f (e) ≥ 0 - see e.g., Chung [54]. 9.1. Flows and cuts 225 space spanned by the characteristic functions of cycles may be identified with the cycle space of G. In fact, these spaces are isomorphic. Under the inner product (9.1), i.e., X (f, g) = f (e)g(e), (9.2) e∈E this vector space is an inner product space. Example 9.1. This example is not needed but is presented for its independent inter- est. Assume G = (V, E) is a strongly connected directed graph. Define the transition probability matrix P for a digraph G by dx , if (x, y) ∈ E, P (x, y) = 0, otherwise, where dx denotes the out-degree. The Perron-Frobenius Theorem states that there exists a unique left eigenvector φ such that (when regarded as a function φ : V → R) φ(v) > 0, P all v ∈ V and φP = ρφ, where ρ is the spectral radius of G. We scale φ so that for v∈V φ(v) = 1. (This vector is sometimes called the Perron vector.) Let Fφ (u, v) = φ(v)P (u, v). Fact: Fφ is a circulation. For a proof, see F. Chung [54]. If the edges of E are indexed in some arbitrary but fixed way then a circulation function restricted to a subgraph H of G may be identified with a vector representation of H, as described above. Thefore, the circulation functions gives a coordinate-free version of the cycle space. The F -vector space spanned by the vector representations of all the segs is called the cocycle space (or the cut space) of G, denoted Z ∗ (G) = Z ∗ (G, F ). This is the column space of the transpose of the incidence matrix of G (§14.1 in Godsil and Royle [87]). It may be regarded as a subspace of C 1 (G, F ) of dimension the rank of G, r(G). When F is a finite field, sometimes the cocycle space is called the cocycle code of G. Lemma 9.2. Under the inner product (9.1) on C 1 (G, F ), the cycle space is orthogonal to the cocycle space. Solution. One proof follows from Theorem 8.3.1 in Godsil and Royle [87]. Here is another proof. By Theorem 2.3 in Bondy and Murty [33, p.27], an edge of G is an edge cut if and only if it is contained in no cycle. Therefore, the vector representation of any cocycle is supported on a set of indices which is disjoint from the support of the vector representation of any cycle. Therefore, there is a basis of the cycle space which is orthogonal to a basis of the cocycle space. Proposition 9.3. Let F = GF (2). The cycle code of a graph G = (V, E) is a linear binary block code of length |E|, dimension equal to the nullity of the graph, n(G), and minimum distance equal to the girth of G. If C ⊂ GF (2)|E| is the cycle code associated to G and C ∗ is the cocycle code associated to G then C ∗ is the dual code of C. In particular, the cocycle code of G is a linear binary block code of length |E|, and dimension r(G) = |E| − n(G). This follows from Hakimi-Bredeson [97] (see also Jungnickel-Vanstone [115]) in the binary case4 . 4 It is likely true in the non-binary case as well, but no proof seems to be in the literature. 226 Chapter 9. Network flows Solution. Let d denote the minimum distance of the code C. Let γ denote the girth of G, i.e., the smallest cardinality of a cycle in G. If K is a cycle in G then the vector vec(K) ∈ GF (2)|E| is an element of the cycle code C ⊂ GF (2)|E| . This implies d ≤ γ. In the other direction, suppose K1 and K2 are cycles in G with associated support vectors v1 = vec(K1 ), v2 = vec(K2 ). Assume that at least one of these cycles is a cycle of minimum length, say K1 , so the weight of its corresponding support vector is equal to the girth γ. The only way that wt(v1 + v2 ) < min{wt(v1 ), wt(v2 )} can occur is if K1 and K2 have some edges in common. In this case, the vector v1 + v2 represents a subgraph which is either a cycle or it is a union of disjoint cycles. In either case, by minimality of K1 , these new cycles must be at least as long. Therefore, d ≥ γ, as desired. Consider a spanning tree T of a graph G and its complementary subgraph T . For each edge e of T the graph T ∪ e contains a unique cycle. The cycles which arise in this way are called the fundamental cycles of G, denoted cyc(T, e). Example 9.4. Consider the graph below, with edges labeled as indicated, together with a spanning tree, depicted to its right, in Figure 9.4. 7 9 0 4 2 6 8 10 1 3 5 Figure 9.1: A graph and a spanning tree for it. You can see from Figure 9.4 that: by adding edge 2 to the tree, you get a cycle 0, 1, 2 with vector representation g1 = (1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0), by adding edge 6 to the tree, you get a cycle 4, 5, 6 with vector representation g2 = (0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0), by adding edge 10 to the tree, you get a cycle 8, 9, 10 with vector representation g3 = (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1). The vectors {g1 , g2 , g3 } form a basis of the cycle space of G over GF (2). The cocycle space of a graph G (also known as the bond space of G or the cut-set space of G) is the F -vector space spanned by the characteristic functions of bonds. Example 9.5. Consider the graph below, with edges labeled as indicated, together with an example of a bond, depicted to its right, in Figure 9.5. You can see from Figure 9.5 that: 9.1. Flows and cuts 227 7 9 0 4 2 6 8 10 1 3 5 Figure 9.2: A graph and a bond of it. by removing edge 3 from the graph, you get a bond with vector representation b1 = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0), by removing edge 7 from the graph, you get a bond with vector representation b2 = (0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0), by removing edges 0, 1 from the graph, you get a bond with vector representation b3 = (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0), by removing edges 1, 2 from the graph, you get a bond with vector representation b4 = (0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0), by removing edges 4, 5 from the graph, you get a bond with vector representation b5 = (0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0), by removing edges 4, 6 from the graph, you get a bond with vector representation b6 = (0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0), by removing edges 8, 9 from the graph, you get a bond with vector representation b7 = (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0), by removing edges 9, 10 from the graph, you get a bond with vector representation b8 = (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1). The vectors {b1 , b2 , b3 , b4 , b5 , b6 , b7 , b8 } form a basis of the cocycle space of G over GF (2). Note that these vectors are orthogonal to the basis vectors of the cycle space in Example 9.4. Note also that the xor sum of two cuts is not a cut.For example, if you xor the bond 4, 5 with the bond 4, 6 then you get the subgraph foormed by the edges 5, 6 and that is not a disconnecting cut of G. 228 Chapter 9. Network flows 9.1.1 Electrical networks We present, as an application of circuit matrices and incidence matrices, and abstract definition of an electrical network. We define an electrical network abstractly as follows. Let G = (V, E) be a simple connected directed graph having n vertices and m edges. We have a current function i : E → R, and a voltage function v : E → R, subject to three conditions. If we index the edges, say as E = {e1 , . . . , em }, then these functions may (and sometimes will) be regarded as column vectors in Rm . The conditions satisfied by the current function are the following. Kirchhoff ’s current law, Ai = ~0, (9.3) where A is the incidence matrix and i is regarded as a column vector. This equation comes from the fact that the algebraic sum of the currents going into a node is zero. Kirchhoff ’s voltage law, Cv = ~0, (9.4) where C is the circuit matrix and v is regarded as a column vector. This equation comes from the fact that the algebraic sum of the voltage drops around a closed loop is zero. In a network with resistors but no inductors or capacitors, there is a relationship between i and v given by Ri = v + b, where b is a vector of external “battery” sources and R is a “resistor matrix.” 9.2 Chip firing games Chip firing games on graphs (which are just pure fun) relate to “abelian sandpile models” from physics to “rotor-routing models” from theoretical computer scientists (designing efficient computer multiprocessor circuits) to “self-organized criticality” (a subdiscipline of dynamical systems) to “algebraic potential theory” on a graph [25] to cryptography (via the Biggs cryptosystem). Moreover, it relates the concepts of the Laplacian of the graph to the tree number to the circulation space of the graph to the incidence matrix, as well as many other ideas. Some good references are [65], [159], [160], [100] and [26]. 9.2. Chip firing games 229 Basic set-up A chip firing game always starts with a directed multigraph G having no loops. A con- figuration is a vertex-weighting, i.e., a function s : V → R. The players are represented by the vertices of G and the vertex-weights represent the number of chips each player (represented by that vertex) has. The initial vertex-weighting is called the starting con- figuration of G. Let vertex v have outgoing degree d+ (v). If the weight of vertex v is ≥ d+ (v) (so that player can afford to give away all their chips) then that vertex is called active. Here is some SAGE/Python code for determining the active vertices. SAGE def active_vertices(G, s): """ Returns the list of active vertices. INPUT: G - a graph s - a configuration (implemented as a list or a dictionary keyed on the vertices of the graph) EXAMPLES: sage: A = matrix([[0,1,1,0,0],[1,0,1,0,0],[1,1,0,1,0],[0,0,0,0,1],[0,0,0,0,0]]) sage: G = Graph(A, format = "adjacency_matrix", weighted = True) sage: s = {0: 3, 1: 1, 2: 0, 3: 1, 4: 1} sage: active_vertices(G, s) [0, 4] """ V = G.vertices() degs = [G.degree(v) for v in V] active = [v for v in V if degs[V.index(v)]<=s[v]] return active If v is active then when you fire v you must also change the configuration. The new configuration s0 will satisfy s0 (v) = s(v) − d+ (v) and s0 (v 0 ) = s(v 0 ) + 1 for each neighbor v 0 of v. In other words, v will give away one chip to each of its d+ (v) neighbors. If x : V → {0, 1}|V | ⊂ R|V | is the representation vector (“characteristic function”) of a vertex then this change can be expressed more compactly as s0 = s − L ◦ x, (9.5) where L is the vertex Laplacian. It turns out that the column sums of L are all 0, so this operation does not change the total number of chips. We use the notation v s → s0 , to indicate that the configuration s0 is the result of firing vertex v in configuration s. Example 9.6. Consider the graph This graph has incidence matrix −1 −1 0 0 0 0 −1 −1 0 0 D= 1 0 −1 −1 0 , 0 0 0 −1 −1 0 0 0 0 1 230 Chapter 9. Network flows 1 • @ @ 0• @•2 4• •3 Figure 9.3: A graph with 5 vertices. and Laplacian 2 −1 0 − 1 0 0 −1 2 −1 0 0 L = D · tD = −1 −1 3 −1 0 . 0 0 −1 2 −1 0 0 0 −1 1 Suppose the initial configuration is s = (3, 1, 0, 1, 1), i.e., player 0 has 3 dollars, player 1 has 1 dollar, player 2 has nothing, player 3 has 1 dollar, player 4 has 1 dollar. Notice player 0 is active. If we fire 0 then we get the new configuration s0 = (1, 2, 1, 1, 1). Indeed, if we compute s0 = s − Lx(0), we get: 3 2 −1 −1 0 0 1 3 2 1 1 −1 2 −1 0 0 0 1 −1 2 s = 0 0 − −1 −1 3 −1 0 0 = 0 − −1 = 1 . 1 0 0 −1 2 −1 0 1 0 1 1 0 0 0 −1 1 0 1 0 1 This can be written more concisely as 0 (3, 1, 0, 1, 1) → (1, 2, 1, 1, 1). We have the cycle 1 0 2 (1, 2, 1, 1, 1) → (2, 0, 2, 1, 1) → (0, 1, 3, 1, 1) → (1, 2, 0, 2, 1) 3 4 → (1, 2, 1, 0, 2) → (1, 2, 1, 1, 1). 9.2. Chip firing games 231 Chip-firing game variants For simplicity, let G = (V, E) be an undirected graph with an indexed set of vertices V = {v1 , . . . , vm } and an indexed set of vertices E = {e1 , . . . , en }. One variant (the “sandpile model”) has a special vertex, called “the sink,” which has special firing properties. In the sandpile variant, the sink is never fired. Another variant (the “dollar game”) has a special vertex, called “the source,” which has special firing properties. In the dollar game variant, the source is only fired when not other vertex is active. We shall consider the dollar game variant here, following Biggs [27]. We select a distinguished vertex q ∈ V , called the “source5 ,” which has a special property to be described below. For the dollar game, a configuration is a function s : V → R for which X s(v) = 0, v∈V and s(v) ≥ 0 for all v ∈ V with v 6= q. A vertex v 6= q can be fired if and only if deg(v) ≤ s(v) (i.e., it “has enough chips”). The equation (9.5) describes the new configuration after firing a vertex. Here is some SAGE/Python code for determining the configuration after firing an active vertex. SAGE def fire(G, s, v0): """ Returns the configuration after firing the active vertex v. INPUT: G - a graph s - a configuration (implemented as a list or a dictionary keyed on the vertices of the graph) v - a vertex of the graph EXAMPLES: sage: A = matrix([[0,1,1,0,0],[1,0,1,0,0],[1,1,0,1,0],[0,0,0,0,1],[0,0,0,0,0]]) sage: G = Graph(A, format = "adjacency_matrix", weighted = True) sage: s = {0: 3, 1: 1, 2: 0, 3: 1, 4: 1} sage: fire(G, s, 0) {0: 1, 1: 2, 2: 1, 3: 1, 4: 1} """ V = G.vertices() j = V.index(v0) s1 = copy(s) if not(v0 in V): raise ValueError, "the last argument must be a vertex of the graph." if not(v0 in active_vertices(G, s)): raise ValueError, "the last argument must be an active vertex of the graph." degs = [G.degree(w) for w in V] for w in V: if w == v0: s1[v0] = s[v0] - degs[j] if w in G.neighbors(v0): s1[w] = s[w]+1 return s1 We say s : V → R is a stable configuration if 0 ≤ s(v) < deg(v), for all v 6= q. The source vertex q can only be fired when no other vertex can be fired, that is only in the case when a stable configuration has been reached. 5 Biggs humorously calls q “the government.” 232 Chapter 9. Network flows Here is some SAGE/Python code for determining the stable vertices. SAGE def stable_vertices(G, s, source = None): """ Returns the list of stable vertices. INPUT: G - a graph s - a configuration (implemented as a list or a dictionary keyed on the vertices of the graph) EXAMPLES: sage: A = matrix([[0,1,1,0,0],[1,0,1,0,0],[1,1,0,1,0],[0,0,0,0,1],[0,0,0,0,0]]) sage: G = Graph(A, format = "adjacency_matrix", weighted = True) sage: s = {0: 3, 1: 1, 2: 0, 3: 1, 4: 1} sage: stable_vertices(G, s) """ V = G.vertices() degs = [G.degree(v) for v in V] if source==None: stable = [v for v in V if degs[V.index(v)]>s[v]] else: stable = [v for v in V if degs[V.index(v)]>s[v] and v!=source] return stable Suppose we are in a configuration s1 . We say a sequence vertices S = (w1 , w2 , . . . , wk ), wi ∈ V not necessarily distinct, is legal if, w1 is active in configuration s1 , for each i with 1 ≤ i < k, si+1 is obtained from si by firing wi in configuration si , for each i with 1 ≤ i < k, wi+1 is active in the configuration si+1 defined in the previous step, the source vertex q occurs in S only if it immediately follows a stable configuration. We call s1 or w1 the start of S. A configuration s is recurrent if there is a legal sequence starting at s which leads back to s. A configuration is critical if it recurrent and stable. Here is some SAGE/Python code for determining a stable vertex resulting from a legal sequence of firings of a given configuration s. I think it returns the unique critical configuration associated to s but have not proven this. SAGE def stabilize(G, s, source, legal_sequence = False): """ Returns the stable configuration of the graph originating from the given configuration s. If legal_sequence = True then the sequence of firings is also returned. By van den Heuvel [1], the number of firings needed to compute a critical configuration is < 3(S+2|E|)|V|ˆ2, where S is the sum of the positive weights in the configuration. EXAMPLES: sage: A = matrix([[0,1,1,0,0],[1,0,1,0,0],[1,1,0,1,0],[0,0,1,0,1],[0,0,0,1,0]]) sage: G = Graph(A, format="weighted_adjacency_matrix") sage: s = {0: 3, 1: 1, 2: 0, 3: 1, 4: -5} sage: stabilize(G, s, 4) {0: 0, 1: 1, 2: 2, 3: 1, 4: -4} REFERENCES: 9.2. Chip firing games 233 [1] J. van den Heuvel, "Algorithmic aspects of a chip-firing game," preprint. """ V = G.vertices() E = G.edges() fire_number = 3*len(V)ˆ2*(sum([s[v] for v in V if s[v]>0])+2*len(E))+len(V) if legal_sequence: seq = [] stab = [] ac = active_vertices(G,s) for i in range(fire_number): if len(ac)>0: s = fire(G,s,ac[0]) if legal_sequence: seq.append(ac[0]) else: stab.append(s) break ac = active_vertices(G,s) if len(stab)==0: raise ValueError, "No stable configuration found." if legal_sequence: return stab[0], seq else: return stab[0] The incidence matrix D and its transpose t D can be regarded as homomorphisms D : C 1 (G, Z) → C 0 (G, Z) and t D : C 0 (G, Z) → C 1 (G, Z). We can also regard the Laplacian L = D · t D as a homomorphism C 0 (G, Z) → C 0 (G, Z). Denote by σ : C 0 (G, Z) → Z the homomorphism defined by X σ(f ) = f (v). v∈V Denote by K(G) the set of critical configurations on a graph G. Lemma 9.7. (Biggs [27]) The set K(G) of critical configurations on a connected graph G is in bijective correspondence with the abelian group Ker(σ)/Im(Q). If you accept this lemma (which we do not prove here) then you must believe that there is a bijection f : K(G) → Ker(σ)/Im(Q). Now, a group operation • on K(G) an be defined by a • b = f −1 (f (a) + f (b)), for all a, b ∈ Ker(σ)/Im(Q). Example 9.8. Consider again the graph 1 • @ @ 0• @•2 4◦ •3 Figure 9.4: A graph with 5 vertices. 234 Chapter 9. Network flows This graph has incidence matrix −1 −1 0 0 0 0 −1 −1 0 0 D= 1 0 −1 −1 0 , 0 0 0 −1 −1 0 0 0 0 1 and Laplacian 2 −1 0 − 1 0 0 −1 2 −1 0 0 L=D· D= t −1 −1 3 −1 0 . 0 0 −1 2 −1 0 0 0 −1 1 Suppose the initial configuration is s = (3, 1, 0, 1, −5), i.e., player 0 has 3 dollars, player 1 has 1 dollar, player 2 has nothing, player 3 has 1 dollar, player 4 is the source vertex q. The legal sequence (0, 1, 0, 2, 1, 0, 3, 2, 1, 0) leads to the stable configuration (0, 1, 2, 1, −4). If q is fired then the configuration (0, 1, 2, 2, −5) is achieved. This is recurrent since it is contained in the cyclic legal sequence 3 2 (0, 1, 2, 2, −5) → (0, 1, 3, 0, −4) → (1, 2, 0, 1, −4) 1 0 q → (2, 0, 1, 1, −4) → (0, 1, 2, 1, −4) → (0, 1, 2, 2, −5). In particular, the configuration (0, 1, 2, 1, −4) is also recurrent. Since it is both stable and recurrent, it is critical. The following result is of basic importance but I’m not sure who proved it first. It is quoted in many of the papers on this topic in one form or another. Theorem 9.9. (Biggs [26], Theorem 3.8) If s is an configuration and G is connected then there is a unique critical configuration s0 which can be obtained by a sequence of legal firings for starting at s. The map defined by the above theorem is denoted γ : C 0 (G, R) → K(G). Another way to define multiplication • on on K(G) is γ(s1 ) • γ(s2 ) = γ(s1 + s2 ), 9.3. Ford-Fulkerson theorem 235 where s1 + s2 is computed using addition on C 0 (G, R). According to Perkinson [159], Theorem 2.16, the critical group satisfies the following isomorphism: K(G) ∼ = Zm−1 /L, where L is the integer lattice generated by the columns of the reduced Laplacian matrix6 . If s is a configuration then we define X wt(s) = s(q) v∈V,v6=q to be the weight of the configuration. The level of the configuration is defined by level(s) = wt(s) − |E| + deg(q). Lemma 9.10. (Merino [144]) If s is a critical configuration then 0 ≤ level(s) ≤ |E| − |V | + 1. This is proven in Theorem 3.4.5 in [144]. What is also proven in [144] is a statement whcih computes the number of critical configurations of a given level in terms of the Tutte polynomial of the associated graph. 9.3 Ford-Fulkerson theorem The Ford-Fulkerson Theorem, or “Max-flow/Min-cut Theorem,” was proven by P. Elias, A. Feinstein, and C.E. Shannon in 1956, and, independently, by L.R. Ford, Jr. and D.R. Fulkerson in the same year. So it should be called the “Elias-Feinstein-Ford-Fulkerson- Shannon Theorem,” to be precise about the authorship. To explain the meaning of this theorem, we need to introduce some notation and terminology. Consider an edge-weighted simple digraph G = (V, E, i, h) without negative weight cycles. Here E ⊂ V (2) , i is an incidence function as in (??), which we regard as the identity function, and h is an orientation function as in (??). Let G be a network, with two distinguished vertices, the “source” and the “sink.” Let s and t denote the source and the sink of G, respectively. The capacity (or edge capacity) ) is a mapping c : E → R, denoted by cuv or c(u, v), for (u, v) ∈ E and h(e) = u. If (u, v) ∈ E and h(e) = v then we set, by convention, c(v, u) = −c(u, v). Thinking of a graph as a network of pipes (representing the edges) transporting water with various junctions (representing vertices), the capacity function represents the maximum amount of “flow” that can pass through an edge. A flow is a mapping f : E → R, denoted by fuv or f (u, v), subject to the following two constraints: f (u, v) ≤ c(u, v), for each (u, v) ∈ V (the “capacity constraint”), P P u∈V, (u,v)∈E f (u, v) = u∈V, (v,u)∈E f (v, u) , for each v ∈ V (conservation of flows). 6 The reduced Laplacian matrix is obtained from the Laplacian matrix by removing the row and column associated to the source vertex. 236 Chapter 9. Network flows An edge (u, v) ∈ E is f -saturated if f (u, v) = c(u, v). An edge (u, v) ∈ E is f -zero if f (u, v) = 0. A path with available capacity is called an “augmenting path.” More precisely, a directed path form s to t is f -augmenting, or f -unsaturated, if no forward edge is f -saturated and no backward edge is f -zero. The value of the flow is defined by X X |f | = f (s, v) − f (v, s), v∈V v∈V where s is the source. It represents the amount of flow passing from the source to the sink. The maximum flow problem is to maximize |f |, that is, to route as much flow as possible from s to t. Example 9.11. Consider the digraph having adjacency matrix 0 1 1 0 0 0 −1 0 −1 1 0 1 −1 1 0 0 1 0 , 0 −1 0 0 0 1 0 0 −1 0 0 1 0 −1 0 −1 −1 0 depicted in Figure 9.5. 2 4 0 1 5 3 Figure 9.5: A digraph with 6 vertices. Suppose that each edge has capacity 1. A maximum flow f is obtained by taking a flow value of 1 along each edge of the path p1 : (0, 1), (1, 5), and a flow value of 1 along each edge of the path p2 : (0, 2), (2, 4), (4, 5). The maximum value of the flow in this case is |f | = 2. This graph can be created in Sage using the commands sage : B = matrix ([[0 ,1 ,1 ,0 ,0 ,0] ,[0 ,0 ,0 ,1 ,0 ,1] ,[0 ,1 ,0 ,0 ,1 ,0] ,[0 ,0 ,0 ,0 ,0 ,1] ,[0 ,0 ,0 ,0 ,0 ,1] ,[0 ,0 , sage : H = DiGraph (B , format = " adjacency_matrix " , weighted = True ) Type H.show(edgewlabels=True) if you want to see the graph with the capacities la- beling the edges. 9.3. Ford-Fulkerson theorem 237 Given a capacitated digraph with capacity c and flow f , we define the residual digraph Gf = (V, E) to be the digraph with capacity cf (u, v) = c(u, v) − f (u, v) and no flow. In other words, Gf is the same graph but it has a different capacity cf and flow 0. This is also called a residual network. Define an s − t cut in our capacitated digraph G to be a partition C = (S, T ) of V such that s ∈ S and t ∈ T . Recall the cut-set of C is the set {(u, v) ∈ E | u ∈ S, v ∈ T }. Lemma 9.12. Let G = (V, E) be a capacitated digraph with capacity c : E → R, and let s and t denote the source and the sink of G, respectively. If C is an s − t cut and if the edges in the cut-set of C are removed, then |f | = 0. Exercise 9.13. Prove Lemma 9.12. The capacity of an s − t cut C = (S, T ) is defined by X c(S, T ) = c(u, v). (s,t)∈(S,T ) The minimum cut problem is to minimize the amount of capacity of an s − t cut. The following theorem is due to P. Elias, A. Feinstein, L.R. Ford, Jr., D.R. Fulkerson, C.E. Shannon. Theorem 9.14. (max-flow min-cut theorem) The maximum value of an s-t flow is equal to the minimum capacity of an s-t cut. See [155] for a simple proof of the max-flow min-cut theorem. The intuitive explana- tion of this result is as follows. Suppose that G = (V, E) is a graph where each edge has capacity 1. Let s ∈ V be the source and t ∈ V be the sink. The maximum flow from s to t is the maximum number of independent paths from s to t. Denote this maximum flow by m. Each s-t cut must intersect each s-t path at least once. In fact, if S is a minimal s-t cut then for each edge e in S there is an s-t path containing e. Therefore, |S| ≤ e. On the other hand, since each edge has unit capacity, the maximum flow value can’t exceed the number of edges separating s from t, so m ≤ |S|. Remark 9.15. Although the notion of an independent path is important for the network- theoretic proof of Menger’s theorem (which we view as a corollary to the Ford-Fulkerson theorem on network flows on networks having capacity 1 on all edges), its significance is less important for networks having arbitrary capacities. One must use caution in generalizing the above intuitive argument to establish a rigorous proof of the general version of the MFMC theorem. Remark 9.16. This theorem can be generalized as follows. In addition to edge capacity, suppose there is capacity at each vertex, that is, a mapping c : V → R, denoted by v 7→ c(v), such that the flow f has to satisfy not only the capacity constraint and the conservation of flows, but also the vertex capacity constraint X f (w, v) ≤ c(v), w∈V 238 Chapter 9. Network flows for each v ∈ V − {s, t}. Define an s − t cut to be the set of vertices and edges such that for any path from s to t, the path contains a member of the cut. In this case, the capacity of the cut is the sum the capacity of each edge and vertex in it. In this new definition, the generalized max-flow min-cut theorem states that the maximum value of an s − t flow is equal to the minimum capacity of an s − t cut.. The idea behind the Ford-Fulkerson algorithm is very simple: As long as there is a path from the source to the sink, with available capacity on all edges in the path, we send as much flow as we can alone along each of these paths. This is done inductively, one path at a time. Algorithm 9.1 Ford-Fulkerson algorithm. Input Graph G = (V, E) with flow capacity c, source s, and sink t. Output A flow f from s to t which is a maximum for all edges in E. 1: f (u, v) ← 0 for each edge uv ∈ E 2: while there is an s-t path p in Gf such that cf (e) > 0 for each edge e ∈ E do 3: find cf (p) = min{cf (u, v) | (u, v) ∈ p} 4: for each edge uv ∈ do 5: f (u, v) = f (u, v) + cf (p) 6: f (v, u) = f (v, u) − cf (p) To prove the max-flow/min-cut theorem we will use the following lemma. Lemma 9.17. Let G = (V, E) be a directed graph with edge capacity c : E → Z, a source s ∈ V , and a sink t ∈ V . A flow f : E → Z is a maximum flow if and only if there is no f -augmenting path in the graph. In other words, a flow f in a capacitated network is a maximum flow if and only if there is no f -augmenting path in the network. Solution. One direction is easy. Suppose that the flow is a maximum. If there is an f -augmenting path then the current flow can be increased using that path, so the flow would not be a maximum. This contradiction proves the “only if” direction. Now, suppose there is no f -augmenting path in the network. Let S be the set of vertices v such that there is an f -unsaturated path from the source s to v. We know s ∈ S and (by hypothesis) t ∈/ S. Thus there is a cut of the form (S, T ) in the network. Let e = (v, w) be any edge in this cut, v ∈ S and w ∈ T . Since there is no f -unsaturated path from s to w, e is f -saturated. Likewise, any edge in the cut (T, S) is f -zero. Therefore, the current flow value is equal to the capacity of the cut (S, T ). Therefore, the current flow is a maximum. We can now prove the max-flow/min-cut theorem. Solution. Let f be a maximum flow. If S = {v ∈ V | there exists an f − saturated path from s to v}, then by the previous lemma, S 6= V . Since T = V − S is non-empty, there is a cut C = (S, T ). Each edge of this cut C in the capacitated network G is f -saturated. 9.3. Ford-Fulkerson theorem 239 Here is some Python code7 which implements this. The class FlowNetwork is basically a Sage Graph class with edge weights and an extra data structure representing the flow on the graph. class Edge : def __init__ ( self ,U ,V , w ): self . source = U self . to = V self . capacity = w def __repr__ ( self ): return str ( self . source ) + " ->" + str ( self . to ) + " : " + str ( self . capacity ) class FlowNetwork ( object ): """ This is a graph structure with edge capacities . EXAMPLES : g = FlowNetwork () map ( g . add_vertex , [ ’ s ’,’o ’,’p ’,’q ’,’r ’,’t ’]) g . add_edge ( ’ s ’,’o ’ ,3) g . add_edge ( ’ s ’,’p ’ ,3) g . add_edge ( ’ o ’,’p ’ ,2) g . add_edge ( ’ o ’,’q ’ ,3) g . add_edge ( ’ p ’,’r ’ ,2) g . add_edge ( ’ r ’,’t ’ ,3) g . add_edge ( ’ q ’,’r ’ ,4) g . add_edge ( ’ q ’,’t ’ ,2) print g . max_flow ( ’ s ’,’t ’) """ def __init__ ( self ): self . adj , self . flow , = {} ,{} def add_vertex ( self , vertex ): self . adj [ vertex ] = [] def get_edges ( self , v ): return self . adj [ v ] def add_edge ( self , u ,v , w =0): assert ( u != v ) edge = Edge (u ,v , w ) redge = Edge (v ,u ,0) edge . redge = redge redge . redge = edge self . adj [ u ]. append ( edge ) self . adj [ v ]. append ( redge ) self . flow [ edge ] = self . flow [ redge ] = 0 def find_path ( self , source , sink , path ): if source == sink : return path for edge in self . get_edges ( source ): residual = edge . capacity - self . flow [ edge ] if residual > 0 and not ( edge , residual ) in path : result = self . find_path ( edge . to , sink , path + [ ( edge , residual ) ]) if result != None : return result def max_flow ( self , source , sink ): path = self . find_path ( source , sink , []) while path != None : flow = min ( res for edge , res in path ) for edge , res in path : self . flow [ edge ] += flow self . flow [ edge . redge ] -= flow path = self . find_path ( source , sink , []) return sum ( self . flow [ edge ] for edge in self . get_edges ( source )) 7 Please see http://en.wikipedia.org/wiki/Ford-Fulkersonwalgorithm. 240 Chapter 9. Network flows 9.4 Edmonds and Karp’s algorithm The objective of this section is to prove Edmond and Karp’s algorithm for the maximum flow-minimum cut problem with polynomial complexity. 9.5 Goldberg and Tarjan’s algorithm The objective of this section is to prove Goldberg and Tarjan’s algorithm for finding maximal flows with polynomial complexity. Chapter 10 Algebraic graph theory 10.1 Laplacian and adjacency matrices 10.2 Eigenvalues and eigenvectors 10.3 Algebraic connectivity 10.4 Graph invariants 10.5 Cycle and cut spaces 10.6 Problems 10.1. A. 241 Chapter 11 Random graphs A random graph can be thought of as being a member from a collection of graphs having some common properties. Recall that Algorithm 2.5 allows for generating a random binary tree having at least one vertex. Fix a positive integer n and let T be a collection of all binary trees on n vertices. It can be infeasible to generate all members of T , so for most purposes we are only interested in randomly generating a member of T . A binary tree of order n generated in this manner is said to be a random graph. This chapter is a digression into the world of random graphs and various models for generating different types of random graphs. Unlike other chapters in this book, our approach is rather informal and not as rigorous as in other chapters. We will discuss some common models of random graphs and a number of their properties without being bogged down in details of proofs. Along the way, we will demonstrate that random graphs can be used to model diverse real-world networks such as social, biological, technological, and information networks. The edited volume [150] provides some historical context for the “new” science of networks. Bollobás [30] and Kolchin [123] provide standard references on the theory of random graphs with rigorous proofs. For comprehensive surveys of random graphs and networks that do not go into too much technical details, see [13, 67, 191, 192]. On the other hand, surveys that cover diverse applications of random graphs and networks and are geared toward the technical aspects of the subject include [4, 29, 47, 58, 59, 63, 154]. 11.1 Network statistics Numerous real-world networks are large, having from thousands up to millions of vertices and edges. Network statistics provide a way to describe properties of networks without concerning ourselves with individual vertices and edges. A network statistic should describe essential properties of the network under consideration, provide a means to differentiate between different classes of networks, and be useful in network algorithms and applications [38]. In this section, we discuss various common network statistics that can be used to describe graphs underlying large networks. 11.1.1 Degree distribution The degree distribution of a graph G = (V, E) quantifies the fraction of vertices in G having a specific degree k. If v is any vertex of G, we denote this fraction by p = Pr[deg(v) = k] (11.1) 242 11.1. Network statistics 243 As indicated by the notation, we can think of (11.1) as the probability that a vertex v ∈ V chosen uniformly at random has degree k. The degree distribution of G is consequently a histogram of the degrees of vertices in G. Figure 11.1 illustrates the degree distribution of the Zachary [203] karate club network. The degree distributions of many real-world networks have the same general curve as depicted in Figure 11.1(b), i.e. a peak at low degrees followed by a tail at higher degrees. See for example the degree distribution of the neural network in Figure 11.2, that of a power grid network in Figure 11.3, and the degree distribution of a scientific co-authorship network in Figure 11.4. 0.3 0.25 0.2 0.15 0.1 5 · 10−2 2 4 6 8 10 12 14 16 (a) Zachary karate club network. (b) Linear scaling. 10−0.6 10−0.8 10−1 10−1.2 10−1.4 100 100.2 100.4 100.6 100.8 101 101.2 (c) Log-log scaling. Figure 11.1: The friendship network within a 34-person karate club. This is more com- monly known as the Zachary [203] karate club network. The network is an undirected, connected, unweighted graph having 34 vertices and 78 edges. The horizontal axis repre- sents degree; the vertical axis represents the probability that a vertex from the network has the corresponding degree. 11.1.2 Distance statistics In chapter 5 we discussed various distance metrics such as radius, diameter, and eccen- tricity. To that distance statistics collection we add the average or characteristic distance d, defined as the arithmetic mean of all distances in a graph. Let G = (V, E) be a simple graph with n = |V | and m = |E|, where G can be either directed or undirected. Then 244 Chapter 11. Random graphs ·10−2 6 10−1.5 4 10−2 2 20 40 60 80 100 120 100 101 102 (a) Linear scaling. (b) Log-log scaling. Figure 11.2: Degree distribution of the neural network of the Caenorhabditis elegans. The network is a directed, not strongly connected, weighted graph with 297 vertices and 2,359 edges. The horizontal axis represents degree; the vertical axis represents the probability that a vertex from the network has the corresponding degree. The degree distribution is derived from dataset by Watts and Strogatz [193] and White et al. [194]. 0.3 10−1 0.2 10−2 0.1 10−3 5 10 15 100 100.2 100.4 100.6 100.8 101 101.2 (a) Linear scaling. (b) Log-log scaling. Figure 11.3: Degree distribution of the Western States Power Grid of the United States. The network is an undirected, connected, unweighted graph with 4,941 vertices and 6,594 edges. The horizontal axis represents degree; the vertical axis represents the probability that a vertex from the network has the corresponding degree. The degree distribution is derived from dataset by Watts and Strogatz [193]. 11.1. Network statistics 245 10−1 0.12 0.1 10−2 8 · 10−2 6 · 10−2 10−3 4 · 10−2 2 · 10−2 10−4 0 0 50 100 150 200 250 100 101 102 (a) Linear scaling. (b) Log-log scaling. Figure 11.4: Degree distribution of the network of co-authorships between scientists posting preprints on the condensed matter eprint archive at http://arxiv.org/archive/ cond-mat. The network is a weighted, disconnected, undirected graph having 40,421 vertices and 175,693 edges. The horizontal axis represents degree; the vertical axis represents the probability that a vertex from the co-authorship network has the corre- sponding degree. The degree distribution is derived from the 2005 update of the dataset by Newman [152]. G has size at most n(n − 1) because for any distinct vertex pair u, v ∈ V we count the edge from u to v and the edge from v to u. The characteristic distance of G is defined by 1 X d(G) = d(u, v) n(n − 1) u6=v∈V where the distance function d is given by ∞, if there is no path from u to v, d(u, v) = 0, if u = v, k, where k is the length of a shortest u-v path. If G is strongly connected (respectively, connected for the undirected case) then our distance function is of the form d : V × V → Z+ ∪ {0}, where the codomain is the set of nonnegative integers. The case where G is not strongly connected (respectively, disconnected for the undirected version) requires special care. One way is to compute the characteristic distance for each component and then find the average of all such characteristic distances. Call the resulting characteristic distance dc , where c means component. Another way is to assign a large number as the distance of non-existing shortest paths. If there is no u-v path, we let d(u, v) = n because n = |V | is larger than the length of any shortest path between connected vertices. The resulting characteristic distance is denoted db , where b means big number. Furthermore denote by dκ the number of pairs (u, v) such that v is not reachable from u. For example, the Zachary [203] karate club network has d = 2.4082 and dκ = 0; the C. elegans neural network [193, 194] has db = 71.544533, dc = 3.991884, and dκ = 20, 268; the Western States Power Grid network [193] has d = 18.989185 and dκ = 0; and the condensed matter co-authorship network [152] has db = 7541.74656, dc = 5.499329, and dκ = 152, 328, 281. 246 Chapter 11. Random graphs We can also define the concept of distance distribution similar to how the degree distribution was defined in section 11.1.1. If ` is a positive integer with u and v being connected vertices in a graph G = (V, E), denote by p = Pr[d(u, v) = `] (11.2) the fraction of ordered pairs of connected vertices in V × V having distance ` between them. As is evident from the above notation, we can think of (11.2) as the probability that a uniformly chosen connected pair (u, v) of vertices in G has distance `. The distance distribution of G is hence a histogram of the distances between pairs of vertices in G. Figure 11.5 illustrates distance distributions of various real-world networks. 0.3 0.4 0.2 0.3 0.2 0.1 0.1 0 1 2 3 4 5 2 4 6 8 10 12 14 (a) Zachary karate club network [203]. (b) C. elegans neural network [193, 194]. ·10−2 0.3 5 4 0.2 3 2 0.1 1 0 0 10 20 30 40 2 4 6 8 10 12 14 16 18 (c) Power grid network [193]. (d) Condensed matter co-authorship net- work [152]. Figure 11.5: Distance distributions for various real-world networks. The horizontal axis represents distance and the vertical axis represents the probability that a uniformly chosen pair of distinct vertices from the network has the corresponding distance between them. 11.2 Binomial random graph model In 1959, Gilbert [85] introduced a random graph model that now bears the name bino- mial (or Bernoulli ) random graph model. First, we fix a positive integer n, a probability 11.2. Binomial random graph model 247 Algorithm 11.1 Generate a random graph in G(n, p). Input Positive integer n and a probability 0 < p < 1. Output A random graph from G(n, p). 1: G ← Kn 2: V ← {0, 1, . . . , n − 1} 3: E ← {2-combinations of V } 4: for each e ∈ E do 5: r ← draw uniformly at random from interval (0, 1) 6: if r < p then 7: add edge e to G 8: return G p, and a vertex set V = {0, 1, . . . , n − 1}. By G(n, p) we mean a probability space over the set of undirected simple graphs on n vertices. If G is any element of the probability space G(n, p) and ij is any edge for distinct i, j ∈ V , then ij occurs as an edge of G independently with probability p. In symbols, for any distinct pair i, j ∈ V we have Pr[ij ∈ E(G)] = p where all such events are mutually independent. Any graph G drawn uniformly at random from G(n, p) is a subgraph of the complete graph Kn and it follows from (1.6) that G has at most n2 edges. Then the probability that G has m edges is given by n pm (1 − p)( 2 )−m . (11.3) Notice the resemblance of (11.3) to the binomial distribution. By G ∈ G(n, p) we mean that G is a random graph of the space G(n, p) and having size distributed as (11.3). To generate a random graph in G(n, p), start with G being a graph on n vertices but G is Kn , the complement of the complete graph on n vertices. no edges. That is, initially Consider each of the n2 possible edges in some order and add it independently to G with probability p. See Algorithm 11.1 for pseudocode of the procedure. The runtime of Algorithm 11.1 depends on an efficient algorithm for generating all 2-combinations of a set of n objects. We could adapt Algorithm ?? to our needs or search for a more efficient algorithm; see problem 11.3 for discussion of an algorithm to generate a graph in G(n, p) in quadratic time. Figure 11.6 illustrates some random graphs from G(25, p) with p = i/6 for i = 0, 1, . . . , 5. See Figure 11.7 for results for graphs in G(2 · 104 , p). The expected number of edges of any G ∈ G(n, p) is n pn(n − 1) α = E[|E|] = p · = 2 2 and the expected total degree is n β = E[# deg] = 2p · = pn(n − 1). 2 Then the expected degree of each edge is p(n − 1). From problem 1.7 we know that the number of undirected simple graphs on n vertices is given by 2n(n−1)/2 248 Chapter 11. Random graphs where (11.3) is the probability of any of these graphs being the output of the above procedure. Let κ(n, m) be the number of graphs from G(n, p) that are connected and have size m, and by Pr[Gκ ] is meant the probability that G ∈ G(n, p) is connected. Apply expression (11.3) to see that (n2 ) X n Pr[Gκ ] = κ(n, i) · pi (1 − p)( 2 )−i i=n−1 where n − 1 is the least number of edges of any undirected connected graph on n vertices, i.e. the size of any spanning tree of a connected graph in G(n, p). Similarly define Pr[κij ] to be the probability that two distinct vertices i, j of G ∈ G(n, p) are connected. Gilbert [85] showed that as n → ∞, then we have Pr[Gκ ] ∼ 1 − n(1 − p)n−1 and Pr[κij ] ∼ 1 − 2(1 − p)n−1 . Algorithm 11.2 Random oriented graph via G(n, p). Input Positive integer n and probability 0 < p < 1. Output A random oriented graph on n vertices. 1: G ← random graph in G(n, p) as per Algorithm 11.3 2: E ← edge set of G 3: G ← directed version of G 4: cutoff ← draw uniformly at random from interval (0, 1) 5: for each edge uv ∈ E do 6: r ← draw uniformly at random from interval (0, 1) 7: if r < cutoff then 8: remove uv from G 9: else 10: remove vu from G 11: return G Example 11.1. Consider a digraph D = (V, E) without self-loops or multiple edges. Then D is said to be oriented if for any distinct pair u, v ∈ V at most one of uv, vu is an edge of D. Provide specific examples of oriented graphs. Solution. If u, v ∈ V is any pair of distinct vertices of an oriented graph D = (V, E), we have various possibilities: 1. uv ∈ / E and vu ∈ / E. 2. uv ∈ E and vu ∈ / E. 3. uv ∈ / E and vu ∈ E. Let n > 0 be the number of vertices in D and let 0 < p < 1. Generate a random oriented graph as follows. First we generate a binomial random graph G ∈ G(n, p) where G is simple and undirected. Then we consider the digraph version of G and proceed to randomly prune either uv or vu from G, for each distinct pair of vertices u, v. Refer to Algorithm 11.2 for pseudocode of our discussion. A Sage implementation follows: 11.2. Binomial random graph model 249 (a) p = 0; α = 0, |E| = 0; β = 0, # deg = 0 (b) p = 1/6; α = 50, |E| = 44; β = 100, # deg = 88 (c) p = 1/3; α = 100, |E| = 108; β = 200, (d) p = 1/2; α = 150, |E| = 156; β = 300, # deg = 212 # deg = 312 (e) p = 2/3; α = 200, |E| = 185; β = 400, (f) p = 5/6; α = 250, |E| = 255; β = 500, # deg = 370 # deg = 510 Figure 11.6: Binomial random graphs G(25, p) for various values of p. 250 Chapter 11. Random graphs ·108 4 α β 3 α̂ β̂ 2 1 0 0 0.2 0.4 0.6 0.8 1 Figure 11.7: Comparison of expected and experimental values of the number of edges and total degree of random simple undirected graphs in G(n, p). The horizontal axis represents probability points; the vertical axis represents the size and total degree (ex- pected or experimental). Fix n = 20, 000 and consider r = 50 probability points chosen as follows. Let pmin = 0.000001, pmax = 0.999999, and F = (pmax /pmin )1/(r−1) . For i = 1, 2, . . . , r = 50 the i-th probability point pi is defined by pi = pmin F i−1 . Each experiment consists in generating M = 500 random graphs from G(n, pi ). For each Gi ∈ G(n, pi ), where i = 1, 2, . . . , 500, compute its actual size αi and actual total degree βi . Then take the mean α̂ of the αi and the mean β̂ of the βi . sage : G = graphs . RandomGNP (20 , 0.1) sage : E = G . edges ( labels = False ) sage : G = G . to_directed () sage : cutoff = 0.5 sage : for u , v in E : ... r = random () ... if r < cutoff : ... G . delete_edge (u , v ) ... else : ... G . delete_edge (v , u ) which produced the random oriented graph in Figure 11.8. Efficient generation of sparse G ∈ G(n, p) The techniques discussed so far (Algorithms 11.1 and 11.9) for generating a random graph from G(n, p) can be unsuitable when the number of vertices n is in the hundreds of thousands or millions. In many applications of G(n, p) we are only interested in sparse random graphs. A linear time algorithm to generate a random sparse graph from G(n, p) is presented by Batagelj and Brandes [18]. The Batagelj-Brandes algorithm for generating a random sparse graph G ∈ G(n, p) uses what is known as a geometric method to skip over certain edges. Fix a probability 0 < p < 1 that an edge will be in the resulting random sparse graph G. If e is an edge of G, we can consider the events leading up to the choice of e as e1 , e2 , . . . , ek where in the i-th trial the event ei is a failure, for 1 ≤ i < k, but the event ek is the first success after k − 1 successive failures. In probabilistic terms, we perform a series of independent trials each having success probability p and stop when the first success 11.2. Binomial random graph model 251 Figure 11.8: A random oriented graph generated using a graph in G(20, 0.1) and cutoff probability 0.5. occurs. Letting X be the number of trials required until the first success occurs, then X is a geometric random variable with parameter p and probability mass function Pr[X = k] = p(1 − p)k−1 (11.4) for integers k ≥ 1, where ∞ X p(1 − p)k−1 = 1. k=1 In other words, waiting times are geometrically distributed. Suppose we want to generate a random number from a geometric distribution, i.e. we want to simulate X such that Pr[X = k] = p(1 − p)k−1 , k = 1, 2, 3, . . . Note that X̀ Pr[X = k] = 1 − Pr[X > ` − 1] = 1 − (1 − p)`−1 . k=1 In other words, we can simulate a geometric random variable by generating r uniformly at random from the interval (0, 1) and set X to that value of k for which 1 − (1 − p)k−1 < r < 1 − (1 − p)k or equivalently for which (1 − p)k < 1 − r < (1 − p)k−1 252 Chapter 11. Random graphs where 1 − r and r are both uniformly distributed. Thus we can define X by X = min{k | (1 − p)k < 1 − r} ln(1 − r) = min k k > ln(1 − p) ln(1 − r) =1+ . ln(1 − p) That is, we can choose k to be ln(1 − r) k =1+ ln(1 − p) which is used as a basis of Algorithm 11.3. In the latter algorithm, note that the vertex set is V = {0, 1, . . . , n − 1} and candidate edges are generated in lexicographic order. The Batagelj-Brandes Algorithm 11.3 has worst-case runtime O(n + m), where n and m are the order and size, respectively, of the resulting graph. Algorithm 11.3 Linear generation of a random sparse graph in G(n, p). Input Positive integer n and a probability 0 < p < 1. Output A random sparse graph from G(n, p). 1: G ← Kn 2: u ← 1 3: v ← −1 4: while u < n do 5: r ← draw uniformly at random from interval (0, 1) 6: v ← v + 1 + bln(1 − r)/ ln(1 − p)c 7: while v ≥ u and u < n do 8: v ←v−u 9: u←u+1 10: if u < n then 11: add edge uv to G 12: return G Degree distribution Consider a random graph G ∈ G(n, p) and let v be a vertex of G. With probability p, the vertex v is incident with each of the remaining n − 1 vertices in G. Then the probability that v has degree k is given by the binomial distribution n−1 k Pr[deg(v) = k] = p (1 − p)n−1−k (11.5) k and the expected degree of v is E[deg(v)] = p(n − 1). Setting z = p(n − 1), we can express (11.5) as k n−1 n−1 z z Pr[deg(v) = k] = 1− k n−1−z n−1 11.3. Erdős-Rényi model 253 and thus zk Pr[deg(v) = k] → exp(−z) k! as n → ∞. In the limit of large n, the probability that vertex v has degree k approaches the Poisson distribution. That is, as n gets larger and larger any random graph in G(n, p) has a Poisson degree distribution. 11.3 Erdős-Rényi model Let N be a fixed nonnegative integer. The Erdős-Rényi [72, 73] (or uniform) random graph model, denoted G(n, N ), is a probability space over the set of undirected simple graphs on n vertices and exactly N edges. Hence G(n, N ) can be considered as a collection n of ( 2 ) undirected simple graphs on exactly N edges, each such graph being selected with N equal probability. A note of caution is in order here. Numerous papers on random graphs refer to G(n, p) as the Erdős-Rényi random graph model, where in fact this binomial random graph model should be called the Gilbert model in honor of E. N. Gilbert who introduced [85] it in 1959. Whenever a paper makes a reference to the Erdős-Rényi model, one should question whether the paper is referring to G(n, p) or G(n, N ). To generate a graph in G(n, N ), start with G being a graph on n vertices but no n edges. Then choose N of the possible 2 edges independently and uniformly at random and let the chosen edges be the edge set of G. Each graph G ∈ G(n, N ) is associated with a probability n 1 2 N n of being the graph resulting from the above procedure. Furthermore each of the 2 edges has a probability n 1 2 of being chosen. Algorithm 11.4 presents a straightforward translation of the above procedure into pseudocode. Algorithm 11.4 Generation of random graph in G(n, N ). Input Positive integer n and integer N with 0 ≤ N ≤ n2 . Output A random graph from G(n, N ). 1: G ← K nn o 2: E ← e0 , e1 , . . . , e(n)−1 2 3: for i ← 0, 1, . . . , N − 1 do 4: r ← draw uniformly at random from 0, 1, . . . , n2 − 1 5: while er is an edge of G do 6: r ← draw uniformly at random from 0, 1, . . . , n2 − 1 7: add edge er to G 8: return G The runtime of Algorithm 11.4 is probabilistic and can be analyzed via the geometric distribution. If i is the number of edges chosen so far, then the probability of choosing 254 Chapter 11. Random graphs a new edge in the next step is n 2 −i n . 2 We repeatedly choose an edge uniformly at random from the collection of all possible edges, until we come across the first edge that is not already in the graph. The number of trials required until the first new edge is chosen can be modeled using the geometric distribution with probability mass function (11.4). Given a geometric random variable X, we have the expectation ∞ X 1 E[X] = n · p(1 − p)n−1 = . n=1 p Therefore the expected number of trials until a new edge be chosen is n n 2 2 −i from which the expected total runtime is N Z N X n n 2 2 n ≈ n dx i=1 2 − i 0 2 −x n n = · ln n 2 +C 2 2 −N for some constant C. The denominator in the latter fraction becomes zero when n2 = N , which can be prevented by adding one to the denominator. Then we have the expected total runtime ! XN n n n n 2 ∈Θ · ln n 2 i=1 2 − i 2 2 −N +1 which is O(N ) when N ≤ n2 /2, and O(N ln N ) when N = n2 . In other words, Algorithm 11.4 has expected linear runtime when the number N of required edges satisfies N ≤ n2 /2. But for N > n2 /2, we obtain expected linear runtime by generating the n complete graph Kn and randomly delete 2 − N edges from the latter graph. Our discussion is summarized in Algorithm 11.5. Algorithm 11.5 Generation of random graph in G(n, N ) in expected linear time. Input Positive integer n and integer N with 0 ≤ N ≤ n2 . Output A random graph from G(n, N ). n 1: if N ≤ 2 /2 then 2: return result of Algorithm 11.4 3: G ← Kn n 4: for i ← 1, 2, . . . , 2 − N do 5: e ← draw uniformly at random from E(G) 6: remove edge e from G 7: return G 11.4. Small-world networks 255 11.4 Small-world networks Vicky: Hi, Janice. Janice: Hi, Vicky. Vicky: How are you? Janice: Good. Harry: You two know each other? Janice: Yeah, I met Vicky at the mall today. Harry: Well, what a small world! You know, I wonder who else I know knows someone I know that I don’t know knows that person I know. — from the TV series Third Rock from the Sun, season 5, episode 22, 2000. Many real-world networks exhibit the small-world effect: that most pairs of distinct vertices in the network are connected by relatively short path lengths. The small-world effect was empirically demonstrated [146] in a famous 1960s experiment by Stanley Mil- gram, who distributed a number of letters to a random selection of people. Recipients were instructed to deliver the letters to the addressees on condition that letters must be passed to people whom the recipients knew on a first-name basis. Milgram found that on average six steps were required for a letter to reach its target recipient, a number now immortalized in the phrase “six degrees of separation” [93]. Figure 11.9 plots results of an experimental study of the small-world problem as reported in [183]. The small-world effect has been studied and verified for many real-world networks including social: collaboration network of actors in feature films [6,193], scientific publication authorship [48, 92, 151, 152]; information: citation network [165], Roget’s Thesaurus [121], word co-occurrence [62, 76]; technological: internet [51, 75], power grid [193], train routes [171], software [153, 185]; biological: metabolic network [110], protein interactions [109], food web [105, 140], neural network [193, 194]. Watts and Strogatz [190, 191, 193] proposed a network model that produces graphs exhibiting the small-world effect. We will use the notation “” to mean “much greater than”. Let n and k be positive integers such that n k ln n 1 (in particular, 0 < k < n/2) with k being even. Consider a probability 0 < p < 1. Starting from an undirected k-circulant graph G = (V, E) on n vertices, the Watts-Strogatz model proceeds to rewire each edge with probability p. The rewiring procedure, also called edge swapping, works as follows. Let V be uniformly distributed. For each v ∈ V , let e ∈ E be an edge having v as an endpoint. Choose another u ∈ V different from v. With probability p, delete the edge e and add the edge vu. The rewiring must produce a simple graph with the same order and size as G. As p → 1, the graph G goes from k-circulant to exhibiting properties of graphs drawn uniformly from G(n, p). Small-world networks are intermediate between k-circulant and binomial random graphs (see Figure 11.10). The Watts-Strogatz model is said to provide a procedure for interpolating between the latter two types of graphs. 256 Chapter 11. Random graphs 15 10 frequency 5 0 0 2 4 6 8 10 number of intermediaries Figure 11.9: Frequency distribution of the number of intermediaries required for letters to reach their intended addressees. The distribution has a mean of 5.3, interpreted as the average number of intermediaries required for a letter to reach its intended destination. The plot is derived from data reported in [183]. (a) p = 0, k-circulant (b) p = 0.3, small-world (c) p = 1, random Figure 11.10: With increasing randomness, k-circulant graphs evolve to exhibit prop- erties of random graphs in G(n, p). Small-world networks are intermediate between k-circulant graphs and random graphs in G(n, p). 11.4. Small-world networks 257 The last paragraph contains an algorithm for rewiring edges of a graph. While the algorithm is simple, in practice it potentially skips over a number of vertices to be considered for rewiring. If G = (V, E) is a k-circulant graph on n vertices and p is the rewiring probability, the candidate vertices to be rewired follow a geometric distribution with parameter p. This geometric trick, essentially the same speed-up technique used by the Batagelj-Brandes Algorithm 11.3, can be used to speed up the rewiring algorithm. To elaborate, suppose G has vertex set V = {0, 1, . . . , n − 1}. If r is chosen uniformly at random from the interval (0, 1), the index of the vertex to be rewired can be obtained from ln(1 − r) 1+ . ln(1 − p) The above geometric method is incorporated into Algorithm 11.6 to generate a Watts- Strogatz network in worst-case runtime O(nk +m), where n and k are as per the input of the algorithm and m is the size of the k-circulant graph on n vertices. Note that lines 7 to 12 are where we avoid self-loops and multiple edges. Algorithm 11.6 Watts-Strogatz network model. Input Positive integer n denoting the number of vertices. Positive even integer k for the degree of each vertex, where n k ln n 1. In particular, k should satisfy 0 < k < n/2. Rewiring probability 0 < p ≤ 1. Output A Watts-Strogatz network on n vertices. 1: M ← nk . sum of all vertex degrees = twice number of edges 2: r ← draw uniformly at random from interval (0, 1) 3: v ← 1 + bln(1 − r)/ ln(1 − p)c 4: E ← contiguous edge list of k-circulant graph on n vertices 5: while v ≤ M do 6: u ← draw uniformly at random from [0, 1, . . . , n − 1] 7: if v − 1 is even then 8: while E[v] = u or (u, E[v]) ∈ E do 9: u ← draw uniformly at random from [0, 1, . . . , n − 1] 10: else 11: while E[v − 2] = u or (E[v − 2], u) ∈ E do 12: u ← draw uniformly at random from [0, 1, . . . , n − 1] 13: E[v − 1] ← u 14: r ← draw uniformly at random from interval (0, 1) 15: v ← v + 1 + bln(1 − r)/ ln(1 − p)c 16: G ← Kn 17: add edges in E to G 18: return G Characteristic path length Watts and Strogatz [193] analyzed the structure of networks generated by Algorithm 11.6 via two quantities: the characteristic path length ` and the clustering coefficient C. The characteristic path length quantifies the average distance between any distinct pair of vertices in a Watts-Strogatz network. The quantity `(G) is thus said to be a global 258 Chapter 11. Random graphs property of G. Watts and Strogatz characterized as small-world those networks that exhibit high clustering coefficients and low characteristic path lengths. Let G = (V, E) be a Watts-Strogatz network as generated by Algorithm 11.6, where the vertex set is V = {0, 1, . . . , n − 1}. For each pair of vertices i, j ∈ V , let dij be the distance from i to j. If there is no path from i to j or i = j, set dij = 0. Thus 0, if there is no path from i to j, dij = 0, if i = j, k, where k is the length of a shortest path from i to j. Since G is undirected, we have dij = dji . Consequently when computing the distance between each distinct pair of vertices, we should avoid double counting by computing dij for i < j. Then the characteristic path length of G is defined by 1 1X `(G) = · dij n(n − 1)/2 2 i6=j X (11.6) 1 = dij n(n − 1) i6=j which is averaged over all possible pairs of distinct vertices, i.e. the number of edges in the complete graph Kn . It is inefficient to compute the characteristic path length via equation (11.6) because we would effectively sum n(n − 1) distance values. As G is undirected, note that 1X X X dij = dij = dij . 2 i6=j i<j i>j The latter equation holds for the following reason. Let D = [dij ] be a matrix of distances for G, where i is the row index, j is the column index, and dij is the distance from i to j. The required sum of distances can be obtained by summing all entries above (or below) the main diagonal of D. Therefore the characteristic path length can be expressed as 2 X `(G) = dij n(n − 1) i<j 2 X = dij n(n − 1) i>j which requires summing n(n−1) 2 distance values. Let G = (V, E) be a Watts-Strogatz network with n = |V |. Set k 0 = k/2, where k is as per Algorithm 11.6. As the rewiring probability p → 0, the average path length tends to n n `→ 0 = . 4k 2k In the special case p = 0, we have n(n + k − 2) `= . 2k(n − 1) ln n However as p → 1, we have ` → ln k . 11.4. Small-world networks 259 Clustering coefficient The clustering coefficient of a simple graph G quantifies the “cliquishness” of vertices in G = (V, E). This quantity is thus said to be a local property of G. Watts and Strogatz [193] defined the clustering coefficient as follows. Suppose n = |V | > 0 and let ni count the number of neighbors of vertex i ∈ V , a quantity that is equivalent to the degree of i, i.e. deg(i) = ni . The complete graph Kni on the ni neighbors of i has ni (ni − 1)/2 edges. The neighbor graph Ni of i is a subgraph of G, consisting of all vertices (6= i) that are adjacent to i and preserving the adjacency relation among those vertices as found in the supergraph G. For example, given the graph in Figure 11.11(a) the neighbor graph of vertex 10 is shown in Figure 11.11(b). The local clustering coefficient Ci of i is the ratio Ni Ci = ni (ni − 1)/2 where Ni counts the number of edges in Ni . In case i has degree deg(i) < 2, we set the local clustering coefficient of i to be zero. Then the clustering coefficient of G is defined by 1X 1X Ni C(G) = Ci = . n i∈V n i∈V ni (ni − 1)/2 3 2 3 4 1 1 5 10 0 6 9 6 7 8 7 8 (a) Graph on 11 vertices. (b) N10 Figure 11.11: The neighbor graph of a vertex. Consider the case where we have a k-circulant graph G = (V, E) on n vertices and a rewiring probability p = 0. That is, we do not rewire any edge of G. Each vertex of G has degree k. Let k 0 = k/2. Then the k neighbors of each vertex in G has 3k 0 (k 0 − 1)/2 edges between them, i.e. each neighbor graph Ni has size 3k 0 (k 0 − 1)/2. Then the clustering coefficient of G is 3(k 0 − 1) . 2(2k 0 − 1) When the rewiring probability is p > 0, Barrat and Weigt [17] showed that the clustering coefficient of any graph G0 in the Watts-Strogatz network model (see Algorithm 11.6) can be approximated by 0 3(k 0 − 1) C(G ) ≈ 0 (1 − p)3 . 2(2k − 1) 260 Chapter 11. Random graphs Degree distribution For a Watts-Strogatz network without rewiring, each vertex has the same degree k. It easily follows that for each vertex v, we have the degree distribution ( 1, if i = k, Pr[deg(v) = i] = 0, otherwise. A rewiring probability p > 0 introduces disorder in the network and broadens the degree distribution, while the expected degree is k. A k-circulant graph on n vertices has nk/2 edges. With the rewiring probability p > 0, a total of pnk/2 edges would be rewired. However note that only one endpoint of an edge is rewired, thus after the rewiring process the degree of any vertex v is deg(v) ≥ k/2. Therefore with k > 2, a Watts-Strogatz network has no isolated vertices. For p > 0, Barrat and Weigt [17] showed that the degree of a vertex v can be written as deg(v) = k/2 + ni with ni ≥ 0, where ni can be divided into two parts α and β as follows. First α ≤ k/2 edges are left intact after the rewiring process, the probability of this occurring is 1 − p for each edge. Second β = ni − α edges have been rewired towards i, each with probability 1/n. The probability distribution of α is k/2 P1 (α) = (1 − p)α pk/2−α α and the probability distribution of β is β pnk/2−β pnk/2 1 1 P2 (β) = 1− β n n where (pk/2)β P2 (β) → exp(−pk/2) β! for large n. Combine the above two factors to obtain the degree distribution min{κ−k/2, k/2} X k/2 (pk/2)κ−k/2−i Pr[deg(v) = κ] = (1 − p)i pk/2−i exp(−pk/2) i=0 i (κ − k/2 − i)! for κ ≥ k/2. 11.5 Scale-free networks The networks covered so far—Gilbert G(n, p) model, Erdős-Rényi G(n, N ) model, Watts- Strogatz small-world model—are static. Once a network is generated from any of these models, the corresponding model does not specify any means for the network to evolve over time. Barabási and Albert [14] proposed a network model based on two ingredients: 1. Growth: at each time step, a new vertex is added to the network and connected to a pre-determined number of existing vertices. 2. Preferential attachment: the newly added vertex is connected to an existing vertex in proportion to the latter’s existing degree. 11.5. Scale-free networks 261 Preferential attachment also goes by the colloquial name of the “rich-get-richer” effect due to the work of Herbert Simon [176]. In sociology, preferential attachment is known as the Matthew effect due to the following verse from the Book of Matthew, chapter 25 verse 29, in the Bible: “For to every one that hath shall be given but from him that hath not, that also which he seemeth to have shall be taken away.” Barabási and Albert observed that many real-world networks exhibit statistical properties of their proposed model. One particularly significant property is that of power-law scaling, hence the Barabási-Albert model is also called a model of scale-free networks. Note that it is only the degree distributions of scale-free networks that are scale-free. In their empirical study of the World Wide Web (WWW) and other real-world networks, Barabási and Albert noted that the probability that a web page increases in popularity is directly proportional to the page’s current popularity. Thinking of a web page as a vertex and the degree of a page as the number of other pages that the current page links to, the degree distribution of the WWW follows a power law function. Power-law scaling has been confirmed for many real-world networks: actor collaboration network [14] citation [60, 165, 170] and co-authorship networks [152] human sexual contacts network [112, 135] the Internet [51, 75, 186] and the WWW [5, 16, 39] metabolic networks [109, 110] telephone call graphs [2, 3] Figure 11.12 illustrates the degree distributions of various real-world networks, plotted on log-log scales. Corresponding distributions for various simulated Barabási-Albert networks are illustrated in Figure 11.13. But how do we generate a scale-free graph as per the description in Barabási and Albert [14]? The original description of the Barabási-Albert model as contained in [14] is rather ambiguous with respect to certain details. First, the whole process is supposed to begin with a small number of vertices. But as the degree of each of these vertices is zero, it is unclear how the network is to grow via preferential attachment from the initial pool of vertices. Second, Barabási and Albert neglected to clearly specify how to select the neighbors for the newly added vertex. The above ambiguities are resolved by Bollobás et al. [32], who gave a precise statement of a random graph process that realizes the Barabási-Albert model. Fix a sequence of vertices v1 , v2 , . . . and consider the case where each newly added vertex is to be connected to m = 1 vertex already in a graph. Inductively define a random graph process (Gt1 )t≥0 as follows, where Gt1 is a digraph on {vi | 1 ≤ i ≤ t}. Start with the null graph G01 or the graph G11 with one vertex and one self-loop. Denote by degG (v) the total (in and out) degree of vertex v in the graph G. For t > 1 construct Gt1 from Gt−11 by adding the vertex vt and a directed edge from vt to vi , where i is randomly chosen with probability ( degGt−1 1 (vs )/(2t − 1), if 1 ≤ s ≤ t − 1, Pr[i = s] = 1/(2t − 1), if s = t. 262 Chapter 11. Random graphs 10−1 10−1 10−2 10−2 −3 10 10−3 10−4 10−4 10−5 10−5 10−6 100 101 102 100 101 102 103 (a) US patent citation network. (b) Google web graph. 10−1 10−2 10−2 10−3 10−3 10−4 10−4 10−5 10−5 10−6 100 101 102 103 104 100 101 102 103 (c) LiveJournal friendship network. (d) Actor collaboration network. Figure 11.12: Degree distributions of various real-world networks on log-log scales. The horizontal axis represents degree and the vertical axis is the corresponding probability of a vertex having that degree. The US patent citation network [133] is a directed graph on 3, 774, 768 vertices and 16, 518, 948 edges. It covers all citations made by patents granted between 1975 and 1999. The Google web graph [134] is a digraph having 875, 713 vertices and 5, 105, 039 edges. This dataset was released in 2002 by Google as part of the Google Programming Contest. The LiveJournal friendship network [10, 134] is a directed graph on 4, 847, 571 vertices and 68, 993, 773 edges. The actor collaboration network [14], based on the Internet Movie Database (IMDb) at http://www.imdb.com, is an undirected graph on 383, 640 vertices and 16, 557, 920 edges. Two actors are connected to each other if they have starred in the same movie. In all of the above degree distributions, self-loops are not taken into account and, where a graph is directed, we only consider the in-degree distribution. 11.5. Scale-free networks 263 10−1 10−1 10−2 10−2 10−3 10−3 10−4 10−4 10−5 10−5 0 10−6 0 10 101 102 103 104 10 101 102 103 104 105 (a) n = 105 vertices (b) n = 106 vertices 10−1 10−1 10−2 10−2 10−3 10−3 10−4 10−4 10−5 10−5 10−6 10−6 10−7 10−7 0 1 2 3 4 5 6 10 10 10 10 10 10 10 100 101 102 103 104 105 106 (c) n = 107 vertices (d) n = 2 · 107 vertices Figure 11.13: Degree distributions of simulated graphs in the classic Barabási-Albert model. The horizontal axis represents degree; the vertical axis is the corresponding probability of a vertex having a particular degree. Each generated graph is directed and has minimum out-degree m = 5. The above degree distributions are only for in-degrees and do not take into account self-loops. 264 Chapter 11. Random graphs The latter process generates a forest. For m > 1 the graph evolves as per the case m = 1; i.e. we add m edges from vt one at a time. This process can result in self- n loops and multiple edges. We write Gm for the collection of all graphs on n vertices n and minimal degree m in the Barabási-Albert model, where a random graph from Gm is n n denoted Gm ∈ Gm . Now consider the problem of translating the above procedure into pseudocode. Fix a positive integer n > 1 for the number of vertices in the scale-free graph to be generated via preferential attachment. Let m ≥ 1 be the number of vertices that each newly added vertex is to be connected to; this is equivalent to the minimum degree that any new vertex will end up possessing. At any time step, let M be the contiguous edge list of all edges created thus far in the above random graph process. It is clear that the frequency (or number of occurrences) of a vertex is equivalent to the vertex’s degree. We can thus use M as a pool to sample in constant time from the degree-skewed distribution. Batagelj and Brandes [18] used the latter observation to construct an algorithm for generating scale- free networks via preferential attachment; pseudocode is presented in Algorithm 11.7. Note that the algorithm has linear runtime O(n + m), where n is the order and m the size of the graph generated by the algorithm. Algorithm 11.7 Scale-free network via preferential attachment. Input Positive integer n > 1 and minimum degree d ≥ 1. Output Scale-free network on n vertices. 1: G ← Kn . vertex set is V = {0, 1, . . . , n − 1} 2: M ← list of length 2nd 3: for v ← 0, 1, . . . , n − 1 do 4: for i ← 0, 1, . . . , d − 1 do 5: M [2(vd + i)] ← v 6: r ← draw uniformly at random from {0, 1, . . . , 2(vd + i)} 7: M [2(vd + i) + 1] ← M [r] 8: add edge (M [2i], M [2i + 1]) to G for i ← 0, 1, . . . , nd − 1 9: return G On the evidence of computer simulation and various real-world networks, it was suggested [14,15] that Pr[deg(v) = k] ∼ k −γ with γ = 2.9 ± 0.1. Letting n be the number of vertices, Bollobás et al. [32] obtained Pr[deg(v) = k] asymptotically for all k ≤ n1/15 and showed as a consequence that γ = 3. In the process of doing so, Bollobás et al. proved various results concerning the expected degree. Denote by #nm (k) the number of vertices of Gnm with in-degree k (and consequently with total degree m + k). For the case m = 1, we have the expectation 1 E[degGt1 (vt )] = 1 + 2t − 1 and for s < t we have 2t E[degGt1 (vs )] = E[degGt−1 (vs )]. 2t − 1 1 Taking the above two equations together, for 1 ≤ s ≤ n we have n Y 2i 4n−s+1 n!2 (2s − 2)! E[degGn1 (vs )] = = . i=s 2i − 1 (2n)!(s − 1)!2 11.6. Problems 265 Furthermore for 0 ≤ k ≤ n1/15 we have 2m(m + 1)n E[#nm (k)] ∼ (k + m)(k + m + 1)(k + m + 2) uniformly in k. As regards the diameter, with n as per Algorithm 11.7, computer simulation by Barabási, Albert, and Jeong [5, 16] and heuristic arguments by Newman et al. [156] suggest that a graph generated by the Barabási-Albert model has diameter approximately ln n. As noted by Bollobás and Riordan [31], the approximation diam(Gnm ) ≈ ln n holds for the case m = 1, but for m ≥ 2 they showed that as n → ∞ then diam(Gnm ) → ln / ln ln n. 11.6 Problems Where should I start? Start from the statement of the problem. What can I do? Visualize the problem as a whole as clearly and as vividly as you can. — G. Polya, from page 33 of [162] 11.1. Algorithm 11.8 presents a procedure to construct a random graph that is simple and undirected; the procedure is adapted from pages 4–7 of Lau [130]. Analyze the time complexity of Algorithm 11.8. Compare and contrast your results with that for Algorithm 11.5. 11.2. Modify Algorithm 11.8 to generate the following random graphs. (a) Simple weighted, undirected graph. (b) Simple digraph. (c) Simple weighted digraph. 11.3. Algorithm 11.1 can be considered as a template for generating random graphs in G(n, p). The procedure does not specify how to generate all the 2-combinations of a set of n > 1 objects. Here we discuss how to construct all such 2-combinations and derive a quadratic time algorithm for generating random graphs in G(n, p). (a) Consider a vertex set V = {0, 1, . . . , n − 1} with at least two elements and let E be the set of all 2-combinations of V , where each 2-combination is written ij. Show that ij ∈ E if and only if i < j. (b) From the previous exercise, we know that if 0 ≤ i < n − 1 then there are n − (i + 1) pairs jk where either i = j or i = k. Show that n−2 X n2 − n (n − i − 1) = i=0 2 and conclude that Algorithm 11.9 has worst-case runtime O((n2 − n)/2). 11.4. Modify the Batagelj-Brandes Algorithm 11.3 to generate the following types of graphs. (a) Directed simple graphs. 266 Chapter 11. Random graphs Algorithm 11.8 Random simple undirected graph. Input Positive integers n and m specifying the order and size, respectively, of the output graph. Output A random simple undirected graph with n vertices and m edges. If m exceeds the size of Kn , then Kn is returned. 1: if n = 1 then 2: return K1 3: max ← n(n − 1)/2 4: if m > max then 5: return Kn 6: G ← null graph 7: A ← n × n adjacency matrix with entries aij 8: aij ← false for 0 ≤ i, j < n 9: i ← 0 10: while i < m do 11: u ← draw uniformly at random from {0, 1, . . . , n − 1} 12: v ← draw uniformly at random from {0, 1, . . . , n − 1} 13: if u = v then 14: continue with next iteration of loop 15: if u > v then 16: swap values of u and v 17: if auv = false then 18: add edge uv to G 19: auv ← true 20: i←i+1 21: return G Algorithm 11.9 Quadratic generation of a random graph in G(n, p). Input Positive integer n and a probability 0 < p < 1. Output A random graph from G(n, p). 1: G ← Kn 2: V ← {0, 1, . . . , n − 1} 3: for i ← 0, 1, . . . , n − 2 do 4: for j ← i + 1, i + 2, . . . , n − 1 do 5: r ← draw uniformly at random from interval (0, 1) 6: if r < p then 7: add edge ij to G 8: return G 11.6. Problems 267 Algorithm 11.10 Briggs’ algorithm for random graph in G(n, N ). Input Positive integers n and N such that 1 ≤ N ≤ n2 . Output A random graph from G(n, N ). n 1: max ← 2 2: if n = 1 or N = max then 3: return Kn 4: G ← Kn 5: u ← 0 6: v ← 1 7: t ← 0 . number of candidates processed so far 8: k ← 0 . number of edges selected so far 9: while true do 10: r ← draw uniformly at random from {0, 1, . . . , max − t} 11: if r < N − k then 12: add edge uv to G 13: k ←k+1 14: if k = N then 15: return G 16: t←t+1 17: v ←v+1 18: if v = n then 19: u←u+1 20: v ←u+1 (b) Directed acyclic graphs. (c) Bipartite graphs. 11.5. Repeat the previous problem for Algorithm 11.5. 11.6. In 2006, Keith M. Briggs provided [37] an algorithm that generates a random graph in G(n, N ), inspired by Knuth’s Algorithm S (Selection sampling technique) as found on page 142 of Knuth [122]. Pseudocode of Briggs’ procedure is presented in Algorithm 11.10. Provide runtime analysis of Algorithm 11.10 and compare your results with those presented in section 11.3. Under which conditions would Briggs’ algorithm be more efficient than Algorithm 11.5? 11.7. Briggs’ Algorithm 11.10 follows the general template of an algorithm that samples without replacement n items from a pool of N candidates. Here 0 < n ≤ N and the size N of the candidate pool is known in advance. However there are situations where the value of N is not known beforehand, and we wish to sample without replacement n items from the candidate pool. What we know is that the candidate pool has enough members to allow us to select n items. Vitter’s algorithm R [187], called reservoir sampling, is suitable for the situation and runs in O(n(1+ln(N/n))) expected time. Describe and provide pseudocode of Vitter’s algorithm, prove its correctness, and provide runtime analysis. 11.8. Repeat Example 11.1 but using each of Algorithms 11.1 and 11.5. 268 Chapter 11. Random graphs 11.9. Diego Garlaschelli introduced [84] in 2009 a weighted version of the G(n, p) model, called the weighted random graph model. Denote by GW (n, p) the weighted random graph model. Provide a description and pseudocode of a procedure to generate a graph in GW (n, p) and analyze the runtime complexity of the algorithm. Describe various statistical physics properties of GW (n, p). 11.10. Latora and Marchiori [129] extended the Watts-Strogatz model to take into ac- count weighted edges. A crucial idea in the Latora-Marchiori model is the concept of network efficiency. Describe the Latora-Marchiori model and provide pseudocode of an algorithm to construct Latora-Marchiori networks. Explain the concepts of local and global efficiencies and how these relate to clustering coefficient and characteristic path length. Compare and contrast the Watts-Strogatz and Latora- Marchiori models. 11.11. The following model for “growing” graphs is known as the CHKNS model [46],1 named for its original proponents. Start with the trivial graph G at time step t = 1. For each subsequent time step t > 1, add a new vertex to G. Furthermore choose two vertices uniformly at random and with probability δ join them by an undirected edge. The newly added edge does not necessarily have the newly added vertex as an endpoint. Denote by dk (t) the expected number of vertices with degree k at time t. Assuming that no self-loops are allowed, show that d0 (t) d0 (t + 1) = d0 (t) + 1 − 2δ t and dk−1 (t) dk (t) dk (t + 1) = dk (t) + 2δ − 2δ . t t As t → ∞, show that the probability that a vertex be chosen twice decreases as t−2 . If v is a vertex chosen uniformly at random, show that (2δ)k Pr[deg(v) = k] = (1 + 2δ)k+1 and conclude that the CHKNS model has an exponential degree distribution. The size of a component counts the number of vertices in the component itself. Let Nk (t) be the expected number of components of size k at time t. Show that N1 (t) N1 (t + 1) = N1 (t) + 1 − 2δ t and for k > 1 show that k−1 ! X iNi (t) (k − i)Nk−i (t) kNk (t) Nk (t + 1) = Nk (t) + δ · − 2δ . i=1 t t t 11.12. Algorithm 11.7 can easily be modified to generate other types of scale-free net- works. Based upon the latter algorithm, Batagelj and Brandes [18] presented a procedure for generating bipartite scale-free networks; see Algorithm 11.11 for 1 Or the “chickens” model, depending on how you pronounce “CHKNS”. 11.6. Problems 269 pseudocode. Analyze the runtime efficiency of Algorithm 11.11. Fix positive inte- ger values for n and d, say n = 10, 000 and d = 4. Use Algorithm 11.11 to generate a bipartite graph with your chosen values for n and d. Plot the degree distribution of the resulting graph using a log-log scale and confirm that the generated graph is scale-free. Algorithm 11.11 Bipartite scale-free network via preferential attachment. Input Positive integer n > 1 and minimum degree d ≥ 1. Output Bipartite scale-free multigraph. Each partition has n vertices and each vertex has minimum degree d. 1: G ← K2n . vertex set is {0, 1, . . . , 2n − 1} 2: M1 ← list of length 2nd 3: M2 ← list of length 2nd 4: for v = 0, 1, . . . , n − 1 do 5: for i = 0, 1, . . . , d − 1 do 6: M1 [2(vd + i)] ← v 7: M2 [2(vd + i)] ← n + v 8: r ← draw uniformly at random from {0, 1, . . . , 2(vd + i)} 9: if r is even then 10: M1 [2(vd + i) + 1] ← M2 [r] 11: else 12: M1 [2(vd + i) + 1] ← M1 [r] 13: r ← draw uniformly at random from {0, 1, . . . , 2(vd + i)} 14: if r is even then 15: M2 [2(vd + i) + 1] ← M1 [r] 16: else 17: M2 [2(vd + i) + 1] ← M2 [r] 18: add edges (M1 [2i], M1 [2i + 1]) and (M2 [2i], M2 [2i + 1]) to G 19: for i = 0, 1, . . . , nd − 1 20: return G 11.13. Find the degree and distance distributions, average path lengths, and clustering coefficients of the following network datasets: (a) actor collaboration [14] (b) co-authorship of condensed matter preprints [152] (c) Google web graph [134] (d) LiveJournal friendship [10, 134] (e) neural network of the C. elegans [193, 194] (f) US patent citation [133] (g) Western States Power Grid of the US [193] (h) Zachary karate club [203] 11.14. Consider the plots of degree distributions in Figures 11.12 and 11.13. Note the noise in the tail of each plot. To smooth the tail, we can use the cumulative degree 270 Chapter 11. Random graphs distribution ∞ X c P (k) = Pr[deg(v) = i]. i=k Given a graph with scale-free degree distribution P (k) ∼ k −α and α > 1, the cumulative degree distribution follows P c (k) ∼ k 1−α . Plot the cumulative degree distribution of each network dataset in Problem 11.13. Appendix A Asymptotic growth Name Standard notation Intuitive notation Meaning theta f (n) = Θ(g(n)) f (n) ∈ Θ(g(n)) f (n) ≈ c · g(n) big oh f (n) = O(g(n)) f (n) ≤ O(g(n)) f (n) ≤ c · g(n) omega f (n) = Ω(g(n)) f (n) ≥ Ω(g(n)) f (n) ≥ c · g(n) little oh f (n) = o(g(n)) f (n) o(g(n)) f (n) g(n) little omega f (n) = ω(g(n)) f (n) ω(g(n)) f (n) g(n) tilde f (n) = Θ̃(g(n)) f (n) ∈ Θ̃(g(n)) f (n) ≈ logΘ(1) g(n) Table A.1: Meaning of asymptotic notations. Class lim f (n)/g(n) = Equivalent definition n→∞ f (n) = Θ(g(n)) a constant f (n) = O(g(n)) and f (n) = Ω(g(n)) f (n) = o(g(n)) zero f (n) = O(g(n)) but f (n) 6= Ω(g(n)) f (n) = ω(g(n)) ∞ f (n) 6= O(g(n)) but f (n) = Ω(g(n)) Table A.2: Asymptotic behavior in the limit of large n. 271 Appendix B GNU Free Documentation License Version 1.3, 3 November 2008 Copyright © 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc. http://www.fsf.org Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The purpose of this License is to make a manual, textbook, or other functional and useful document “free” in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. 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Index A(G), 21 per(G), 185 C 0 (G, F ), 223 rad(G), 185 C 1 (G, F ), 223 , 41 Cn , 16 td, 139 En , 46 ε, 85, 88 Gc , 39 ϕ(n), 61 Kn , 15 f -augmenting, 236 Km,n , 19 f -saturated, 236 Ln , 41 f -unsaturated, 236 Pn , 16, 41 f -zero, 236 Qn , 42 k-coloring, 212 Wn , 35 k-connected, 188 ∆, 35 k-edge-connected, 189 ∆(G), 10 n-queens problem, 145 adj, 4 n-space, 82 L, 27 graph6, 104, 106, 108, 109 L(G), 8 sparse6, 106, 108 Ni , 259 Lukaszewicz, J., 70 χv (G), 213 ∼ =, 28 edge expander family, 204 deg, 5, 8 deg+ , 7 active vertex, 229 deg− , 7 acyclic, 51, 63, 96 δ(G), 10 adjacency matrix, 21 depth(v), 52 reduced, 23 diam(G), 185 signed edge, 199 , 183 algorithm , 255 greedy, 63, 66, 126, 127 height(T ), 52 optimization, 57 iadj, 5 random, 48, 96–98, 148, 247, 248, 252– id, 5 254, 257, 264, 266, 267, 269 κ(G), 188 recursive, 101 κe (G), 189 alphabet, 79, 80, 85 κv (G), 187 binary, 85, 86 λ(G), 189 English, 79, 98 lg, 154 weighted, 80, 87 oadj, 5 Altito, Noelie, 135 od, 5 arcs, 3 ω, 14 ASCII, 79, 81, 106, 108 ω(G), 213 augmenting path, 236 G, 39 Australian National University, 106 288 Index 289 automata theory, 134 properties, 164 AVL tree, 181 root-degree property, 164, 165 biology, 182 backtrack, 116 bipartite graph, 16, 20, 48, 268, 269 algorithm, 145 complete, 19, 20 Baker, Matthew, 183 Birkhoff-Lewis Conjecture, 221 balanced bracket problem, 142, 143 bit, 79, 85, 108 Barabási-Albert model, 261 least significant, 108 Batagelj, Vladimir, 250, 268 most significant, 108 Batagelj-Brandes algorithm, 250, 252 parity, 108 Baudot, E., 81 bit vector, 106, 108, 109 Bellman, Richard E., 127 length, 108 Bellman-Ford algorithm, 123, 127–130, 135 bond, 38, 57, 224 Benjamin, Arthur T., 102 Borůvka bent, 200 algorithm, 63, 70, 71, 95, 96, 101 Bernoulli family, 53 Otakar, 63, 70 BFS, 110–114, 116, 120 bowtie graph, 13 big-endian, 108, 109 braille, 80 Biggs, Norman, 87 branch cut, 54, 56 binary heap, 150, 151 Brandes, Ulrik, 250, 268 maximum, 178 breadth-first search, 56, 90, 91, 110–114, minimum, 178 119, 120, 122–124, 137, 141 order property, 152, 180 tree, 110, 113 sift-down, 157 bridge, 38, 52, 58, 70, 74, 189 sift-up, 155 bridgeless, 189 structure property, 152 Briggs binary search, 138, 169 algorithm, 267 binary search tree, 150, 169, 171, 178 Keith M., 267 left subtree property, 169 Brooks’ inequality, 215 property, 169, 180 BST, 169 recursion property, 169 bubble sort, 140, 141 right subtree property, 169 butterfly graph, 13 binary tree, 54, 56, 75, 77–79, 98, 150 complete, 77, 78 Caenorhabditis elegans, 244 nearly complete, 152 canonical label, 30 random, 80, 98 Cantor-Schröder-Bernstein theorem, 48 Binet capacity, 235 formula, 181 cut, 237 Jacques Philippe Marie, 181 card, 116 binomial cardinality, 8 coefficient, 162, 178 Carroll, Lewis, 2 distribution, 247 Cartesian product, 41 random graph, 249 Catalan tree, 162 number, 46, 47, 78 binomial heap, 150, 161, 162, 164, 179 recursion, 78 maximum, 179 Cayley graph of Boolean function, 199 minimum, 179 characteristic polynomial, 198 order property, 164, 180 Chazelle, Bernard, 179 290 Index check matrix, 23 optimal, 87 chemistry, 133, 182 prefix, 80 chess, 80, 114, 145 prefix-free, 80, 81, 86, 98 chessboard, 114 radix, 101 knight, 114 reliability, 80 knight piece, 114 security, 80 knight’s tour, 114–116, 145, 146 tree representation, 85, 86 queen, 145, 146 uniquely decodable, 86 child variable-length, 80 left, 75, 77, 90, 93 codeword, 79, 85 right, 75, 90, 93 length, 87 Chinese ring puzzle, 81, 82 coding function, 80 chip firing game, 229 Cohen, Danny, 108 chip-firing game, 229 Collatz chip-firing games, 228 conjecture, 99 CHKNS model, 268 graph, 99, 100 Choquet, G., 70 length, 99 chromatic sequence, 99 index, 215 tree, 99, 100 number, 213 color code, 80 polynomial, 220 color-class, 213 root, 221 combinatorial generation, 178 chromatically equivalent, 220 combinatorial graphs, 2 Chu Shi-Chieh, 179 combinatorics, 82 Chvátal graph, 96, 97 communications network, 191 circuit, 13 complement, 39 board, 110 complete graph, 15, 16, 97, 98, 247, 252, electronic, 62 254, 257, 258, 266, 267 circuit matrix, 73 component, 14, 34, 58 circulation, 224 connected, 63 circulation space, 224, 225 computer science, 90 classification tree, 52, 53, 57 condensed matter, 245 claw graph, 188 configuration, 229, 231 clique, 213 critical, 232 maximal, 213 level, 235 number, 213 recurrent, 232 closed form, 78 stable, 231 closeness centrality, 196 starting, 229 cocycle code, 225 weight, 235 cocycle space, 225 connected graph, 14, 57 cocyle, 224 connectivity, 141 code, 79, 85 cost, 121 r-ary, 101 Coward, Noel, 11 binary, 79, 85 critical configurations, 233 block, 80 critical group, 233, 235 economy, 80 cryptosystem, 80 error-correcting, 23, 80 current, 228 linear, 82 cut Index 291 set, 38, 59 distance, 52, 103, 113, 114, 121, 122, 124, cut set, 224 126, 128, 130 cut space, 225 characteristic, 245 cut-edge, 188, 189 function, 121, 122, 182, 183, 205 cut-point, 187 matrix, 28, 122 cut-vertex, 187, 188 minimum, 124 cutset, 74 total, 139 matrix, 74 distance distribution, 246 cycle, 13, 51, 52, 58, 60, 122, 123, 136 distribution fundamental, 60, 98 binomial, 252 negative, 122, 123, 128, 130, 135, 136, geometric, 251, 253 148 Poisson, 253 cycle code, 224 uniform, 252 cycle double cover conjecture, 189 divide and conquer, 148 cycle graph, 16, 17, 49 dollar game, 231 cycle matrix, 73 Dryden, John, 110 cycle space, 25, 224 dynamic programming, 130 binary, 25 eccentricity, 183, 184 D’Angelo, Anthony J., 95 mutual, 206 Dörrie, Heinrich, 47 path, 205 data structure, 103, 149 vertex, 206 de Moivre, Abraham, 181 edge, 3 de Montmort, Pierre Rémond, 145 boundary, 204 decode, 79 capacity, 235 degree, 5, 8 contraction, 39 matrix, 27 cut, 38, 57, 193 maximum, 10, 56 deletion, 38 minimum, 10, 61 deletion subgraph, 36 sequence, 30, 61 directed, 4 weighted, 8 endpoint, 63 degree centrality, 196 expansion, 204 degree distribution, 242–245, 252, 262, 263 head, 6 depth-first search, 56, 92, 110, 114, 116– incident, 3 120, 123, 137, 141 multigraph, 6 tree, 116, 119 multiple, 3 de Moivre, Abraham, 145 tagging game, 57 DFS, 114, 116–120 tail, 6 diameter, 113, 114 weight, 6 Digital Signature Algorithm, 101 edge chromatic number, 215 digraph, 5, 52 edge coloring, 215 weighted, 121 edge cut subgraph, 224 Dijkstra edge cutset, 224 algorithm, 14, 124–128, 135, 149 edge-cut, 188 E. W., 66, 124 Edmonds, Jack, 57 Dirac’s theorem, 195 eigenvalue, 101 disconnected graph, 14 electrical network, 228 disconnecting set, 38 element 292 Index random, 79 frequency distribution, 256 Elkies, Noam D., 116 friendship graph, 197 encode, 79 FRW, 128, 130 endianness, 108 function plot, 2 England, 147 fundamental cycles, 226 entropy encoding, 80 Gallai, Tibor, 31 function, 80 Garlaschelli, Diego, 268 Erdős, Paul, 31 genetic code, 80 error rate, 80 Gilbert, E. N., 253 Euler girth, 13 Leonhard, 1, 10, 46, 61 Goldbach, Christian, 46 phi function, 61, 98 Goldberg, R., 128 phi sequence, 61, 62 golden ratio, 102, 181 polygon division problem, 46 Graham, Ronald L., 187 subgraph, 13 graph, 3 Euler subgraph, 224 connected, 14, 120, 121 Eulerian trail, 1 cube, 76 expander graph, 204 dense, 105, 130 Desargues, 76, 77 Faber, Xander, 183 directed, 5 family tree, 14, 15, 52, 53 disconnected, 14 fault-tolerant, 191 Dyck, 213 Fermat’s little theorem, 48 Foster, 18 Fibonacci Franklin, 17 sequence, 181 Frucht, 215 tree, 181 Gray, 20 FIFO, 111, 116 Heawood, 217 filesystem, 52 icosahedral, 218 hierarchy, 52 intersection, 34, 35 finite automaton, 49 join, 35 first in, first out, 111 line, 8 flag semaphore, 80 nonisomorphic, 46 Florek, K., 70 Pappas, 219 flow, 224, 235 simple, 8 value, 236 sparse, 21, 105, 130, 135 flow chart, 103 traversal, 110 flow space, 224 trivial, 61, 70 Floyd, Robert, 130 undirected, 3 Floyd-Roy-Warshall algorithm, 128, 130– union, 34, 35 132, 134, 186, 187 unweighted, 3 football, 80 weighted, 6, 63, 66, 121 forbidden minor, 43 graph isomorphism, 28, 30 Ford, Lester Randolph, Jr., 127 graph minor, 43 forest, 51, 52 graphical sequence, 31, 33 Franklin graph, 29 Gray code, 81, 82 Frederickson, Greg N., 179 m-ary, 81 FreeBSD, 103 binary, 81, 82 Index 293 reflected, 82, 84 unoriented, 24 Gray, Frank, 81 indegree, 5 Gribkovskaia, Irina, 1 unweighted, 7 grid, 42 independent set, 213 graph, 52, 54, 65, 66, 96, 98, 148 induction, 58, 61, 77, 86, 87, 96 Gros, L., 81 structural, 96 group theory infix notation, 142 computational, 82 information channel, 80 Gulliver’s Travels, 108 Internet, 261 topology, 262 Hadamard transform, 200 invariant, 29, 33 Hakimi, S. L., 31 isomorphism, 61 Halskau Sr., Øyvind, 1 isoperimetric number, 204 Hamming distance, 42 Hampton Court Palace, 147 Jarnı́k, V., 66 handshaking lemma, 10 Johnson Havel, Václav, 31 algorithm, 123, 135 Havel-Hakimi Donald B., 135 test, 33 join, 60 theorem, 31 Jordan, Camille, 186 heap 2-heap, 66 Königsberg, 1 k-ary, 127 graph, 2, 6 binary, 127 seven bridges puzzle, 1, 10 binary minimum, 88 Kaliningrad, 1 Fibonacci, 66, 127, 135 Kaplan, Haim, 179 heapsort, 151 Kataoka, Seiji, 95 Heawood graph, 215 Kinoshita, Harunobu, 148 hierarchical structure, 14, 51, 52 Kirchhoff’s current law, 228 Hopcroft, John E., 114 Kirchhoff’s voltage law, 228 Hopkins, Brian, 1 Kleene house graph, 3 algorithm, 134 Huffman Stephen, 134 David, 87 Klein, Felix, 2 tree, 149 Kneser graph, 104–106 Huffman code, 85, 87–89, 91, 98 Knuth binary, 87 Algorithm S, 267 encoding, 90 Donald E., 81, 267 tree construction, 87 Kraft tree representation, 88, 89, 91 inequality, 101, 102 Humpty Dumpty, 2 Leon Gordon, 102 hypercube graph, 42, 82, 83 theorem, 102 Kruskal in-neighbor, 5 algorithm, 63–66, 95, 96, 101 incidence Joseph B., 63 function, 6 matrix, 24 ladder graph, 41 incidence matrix Lagarias, Jeffrey C., 99 oriented, 24 Laplacian, 233 294 Index reduced, 235 McMillan Laplacian matrix, 27, 101, 198 Brockway, 101 Laporte, Gilbert, 1 theorem, 101 last in, first out, 116 Menger’s theorem, 192–194 Latora, V., 268 merge sort, 166 Latora-Marchiori model, 268 Merkle, Ralph C., 101 lattice, 42 Merris-McKay theorem, 99 Lee, C. Y., 110 mesh, 42, 43 legal firing sequence, 232 message, 85 Lehman, A., 57 metabolic network, 261 Lehmer, D. H., 47 metric, 122, 183 level function, 121 binary tree, 152 metric graph, 183 tree, 180 metric space, 122 LIFO, 116 finite, 122 line graph, 8 Milgram, Stanley, 255 linear search, 138 minimum cut problem, 237 Linux, 52 minimum spanning tree problem, 63 list, 88, 104, 111, 113, 116, 127 molecular graph, 132 adjacency, 27, 104, 105, 113 Montmort-Moivre strategy, 145 contiguous edge, 106, 257 Moore, Edward F., 110, 127 edge, 106 Morse code, 81, 86, 98 element, 104 MST, 63 empty, 104 multi-undirected graph, 5 length, 104 multidigraph, 5 little-endian, 108 multigraph, 5 Loberman, H., 63 adjacency, 7 Loebbing, Martin, 116 in-neighbor, 7 Lucas out-neighbor, 7 M. Édouard, 102, 114 Munroe, Randall, 51, 103, 114, 124, 127, number, 102 149, 210, 211 musical score, 80 Marchiori, M., 268 neighbor graph, 259 matrix, 21 network, 235 adjacency, 22, 104, 105, 109 biological, 244, 255 bi-adjacency, 23 citation, 262 cutset, 76 collaboration, 262 distance, 187 communication, 57 incidence, 26 information, 255 main diagonal, 109 social, 245, 255, 262 transpose, 45 technological, 244, 255 upper triangle, 109 Zachary karate club, 243 Matthew effect, 261 node, 3 max-flow min-cut theorem, 237 noisy channel, 80 generalized, 238 null graph, 4, 261 maximum flow problem, 236 Nuutila, Esko, 134 maze, 110, 114, 147 McKay, Brendan D., 106, 116 order, 3 Index 295 organism, 53, 57 Pregel River, 1 orientation, 6, 24 Prim oriented graph, 248 algorithm, 63, 66–69, 95, 96, 101, 149 random, 248, 251 R. C., 63, 66 out-neighbor, 5, 111, 119, 123, 124 priority queue, 149, 150 outdegree, 5 probability, 87 unweighted, 7, 8 expectation, 87 outer boundary, 204 sample space, 79 overfull graph, 45 space, 247 Oxley, James, 57 pseudorandom number, 47, 251 Python, 21 Paley graph, 205 parallel forest-merging, 70 queue, 92, 111, 113, 116, 120, 124 parallelization, 70 dequeue, 91, 93, 111, 113 partition, 48 end, 111 Pascal enqueue, 91, 93, 111, 113 formula, 164, 178 front, 111 path, 12, 13, 51, 52 length, 111 closed, 13 minimum-priority, 66, 88, 135 distance, 121 priority, 88 even, 13 rear, 111 start, 111 geodesic, 14 graph, 41 Ramanujan graph, 204 Hamiltonian, 82, 83 random graph, 242 internally disjoint, 191 Bernoulli, 246 length, 52, 121, 122 binomial, 246, 255 odd, 13 Erdős-Rényi, 253 shortest, 103, 121–124, 126, 128, 130, uniform, 253 135 weighted, 268 tree, 59, 60 random variable weighted, 135 geometric, 251 path graph, 16 Rasmussen, Rune, 57 pendant, 8, 60 recurrence relation, 181 perfect square, 53 recursion, 57, 58, 70, 71, 87, 93, 96, 130 Perkal, J., 70 reduced Laplacian, 235 permutation regular graph, 10, 48 equivalent, 30 k-circulant, 49, 255, 257 random, 98 r-regular, 10, 49 Petersen relative complement, 39 graph, 119, 120, 188, 189 remainder, 108 Julius, 119 reservoir sampling, 267 planar graph, 47 residual digraph, 237 plane, 148 residual network, 237 Pollak, O., 187 reverse Polish notation, 142 postfix notation, 142 rich-get-richer effect, 261 power grid, 244 river crossing problem, 141 preferential attachment, 260, 261, 264 Robertson, Neil, 43 prefix-free condition, 80 Robertson-Seymour theorem, 43 296 Index Roget’s Thesaurus, 255 spanning forest, 70 root directory, 52 spanning subgraph, 15 root list, 165 spanning tree, 52, 54, 57, 62–64, 96, 98, 102 rotor-routing model, 228 maximum, 95 Roy, Bernard, 130 minimum, 62–64, 66–71, 96 RSA, 101 randomized construction, 96, 97, 101 Runde, Volker, 122 sparse graph, 250 Russia, 1 spectrum, 198 Laplacian, 198 sandpile model, 231 stack, 92, 116, 119, 120 saturated edge, 236 length, 116 scale-free network, 264, 268, 269 pop, 92, 93, 95, 116, 119 scatterplot, 2, 62, 84 push, 92, 93, 95, 116, 119 Schulz, Charles M., 176 Stanley, Richard P., 116 scientific collaboration, 245 star graph, 19, 20 seg, 224 state selection sort, 140, 141 final, 141 self-complementary graph, 39 initial, 141 self-loop, 4 Steinhaus, H., 70 separating set, 38, 193 string, 79, 85 set, 104 empty, 88 n-set, 2 Strogatz, Steven H., 255 totally ordered, 150 strongly regular graph, 200 Seymour subgraph, 11, 15 Paul, 43, 189 edge-deletion, 62, 63, 99 Shannon subtree, 57, 93 Claude E., 8, 57 left, 77, 93 multigraphs, 8, 9 right, 77, 93 switching game, 57, 58 supergraph, 15 Shimbel, A., 127 Swift, Jonathan, 108 Shirali, Satish, 122 Sylvester’s Law of Nullity, 73 shortest path, 14 symbol, 85 Simon, Herbert, 261 symbolic computation, 142 simple graph, 8, 98, 255 symmetric difference, 35 random, 247, 252–254, 266, 267 Szekeres, G., 189 single-source shortest path, 124, 127 six degrees of separation, 255 Takaoka, Tadao, 178, 179 size, 3 Tanner graph, 23 component, 268 Tarjan, Robert Endre, 114 tree, 58–60 Tarry, Gaston, 114 small-world, 49, 114, 257 telegraph, 81 algorithm, 257 The Brain puzzle, 82 characteristic path length, 257 Thoreau, Henry David, 8 clustering coefficient, 257, 259 Through the Looking Glass, 2 effect, 255 topology, 122 experimental results, 256 total order, 150 network, 257 Tower of Hanoi puzzle, 82 social network, 182 trail, 12, 13 Index 297 closed, 13 deletion, 37, 93 transition probability matrix, 225 deletion subgraph, 35, 187 transitive closure, 133, 134 endpoint, 51, 52 trapdoor function, 80 expansion, 204 traveling salesman problem, 103 head, 4 traversal internal, 52, 191 bottom-up, 93, 94, 101 isolated, 8, 57, 104 in-order, 93, 94, 101, 169 isoperimetric number, 204 level-order, 90, 91, 93, 101 leaf, 51, 52, 85, 93 post-order, 92, 93, 101 multigraph, 6 pre-order, 92, 101 parent, 52 treasure map, 1 root, 15, 52, 54, 56, 85 tree, 14, 51, 52, 60, 110 set, 3 2-ary, 75 source, 124, 136 n-ary, 52 tail, 4 binary, 52, 85, 88, 93 union, 34 complete, 52, 57 vertex connectivity, 187 depth, 52 vertex-cut, 187 directed, 52 Vijay, Sujith, 31 expression, 52, 53 Vitter height, 52 algorithm, 267 nonisomorphic, 52, 54, 55 Jeffrey Scott, 267 ordered, 52, 90 Vizing’s theorem, 218 recursive definition, 57, 58, 96 voltage, 228 rooted, 15, 52, 56, 75, 110, 119 Vuillemin, Jean, 164 subtree, 58 traversal, 90, 92 Wagner triangle inequality, 122, 136, 183 conjecture, 43 Tripathi, Amitabha, 31 Klaus, 43 trivial graph, 16, 52, 268 walk, 11, 13 tuple, 2 closed, 13 length, 11, 12 union trivial, 11 digraph, 61, 62 Walsh transform, 200 union-find, 96 Warshall, Stephen, 130, 134 Unix, 52 Watanabe, Kohtaro, 95 unweighted degree, 7 Watts, Duncan J., 255 USA, 244, 262 Watts-Strogatz model, 255, 257, 268 value of flow, 236 Wegener, Ingo, 116 Vandermonde weight, 63, 121, 122, 130 Alexandre-Théophile, 179 correcting, 122, 123 convolution, 179 function, 63, 122, 136 Vasudeva, Harkrishan L., 122 graph, 6 vertex, 3 minimum, 63, 66, 70, 126 adjacent, 3 multigraph, 6 child, 52, 56 negative, 122, 123, 127, 128 cut, 38, 193 nonnegative, 121–124, 126, 135 degree, 10 path, 182 298 Index positive, 122 reweight, 135, 136 setting, 122, 123 unit, 121, 122 Weinberger, A., 63 wheel graph, 35, 102 Whitney Hassler, 191 inequality, 190 theorem, 195 Wiener Harold, 99, 132 number, 99, 101, 132 Williams, J. W. J., 151 Wilson, Robin, 1 wine, 197, 202, 203, 207, 208 word, 85 World Wide Web, 261 Yamada, Takeo, 95, 148 Yerger, Carl R., 102 Zachary, Wayne W., 243 zero padding, 108 Zubrzycki, S., 70 Zwick, Uri, 179