Open Data Structures (in C++) Pat Morin ∆ Edition 0.1 Acknowledgments I am grateful to Nima Hoda, who spent a summer tirelessly proofreading many of the chapters in this book, and to the students in the Fall 2011 offering of COMP2402/2002, who put up with the first draft of this book and spotted many typographic, grammatical, and factual errors in the first draft. i Preface to the C++ Edition This book is intended to teach the design and analysis of basic data structures and their implementation in an object-oriented language. In this edition, the language happens to be C++. This book is not intended to act as an introduction to the C++ programming language. Readers of this book need only be familiar with the basic syntax of C++ and similar languages. Those wishing to work with the accompanying source code should have some experience programming in C++. This book is also not intended as an introduction to the C++ Standard Template Library or the generic programming paradigm that the STL embodies. This book describes implementations of several different data structures, many of which are used in implemen- tations of the STL. The contents of this book may help an STL programmer understand how some of the STL data structures are implemented and why these implementations are efficient. iii Why This Book? There are plenty of books that teach introductory data structures. Some of them are very good. Most of them cost money, and the vast majority of computer science undergraduate students will shell-out at least some cash on a data structures book. There are a few free data structures books available online. Some are very good, but most of them are getting old. The majority of these books became free when the author and/or publisher decided to stop updating them. Updating these books is usually not possible, for two reasons: (1) The copyright belongs to the author or publisher, who may not allow it. (2) The source code for these books is often not available. That is, the Word, WordPerfect, FrameMaker, or LATEX source for the book is not available, and the version of the software that handles this source may not even be available. The goal of this project is to forever free undergraduate computer science students from having to pay for an introductory data structures book. I have decided to implement this goal by treating this book like an Open Source software project. The LATEX source, C++ source, and build scripts for the book are available for download on the book’s website (opendatastructures.org) and also — more importantly — on a reliable source code management site (https://github.com/patmorin/ods). This source code is released under a Creative Commons Attribution license, meaning that anyone is free • to Share — to copy, distribute and transmit the work; and • to Remix — to adapt the work. This includes the right to make commercial use of the work. The only condition on these rights is • Attribution — You must attribute the work by displaying a notice stating the derived work contains code and/or text from the Open Data Structures Project and/or linking to opendatastructures.org. v Anyone can contribute corrections/fixes using the git source-code management sys- tem. Anyone can fork from the current version of the book and develop their own version (for example, in another programming language). They can then ask that their changes be merged back into my version. My hope is that, by doing things this way, this book will continue to be a useful textbook long after my interest in the project (or my pulse, whichever comes first) has waned. vi Contents 1 Introduction 1 1.1 Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Queue, Stack, and Deque Interfaces . . . . . . . . . . . . . . . . 2 1.1.2 The List Interface: Linear Sequences . . . . . . . . . . . . . . . . . 2 1.1.3 The USet Interface: Unordered Sets . . . . . . . . . . . . . . . . . . 3 1.1.4 The SSet Interface: Sorted Sets . . . . . . . . . . . . . . . . . . . . . 4 1.2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Asymptotic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.4 Randomization and Probability . . . . . . . . . . . . . . . . . . . . . 8 1.3 The Model of Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Code Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5 List of Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Array-Based Lists 15 2.1 ArrayStack: Fast Stack Operations Using an Array . . . . . . . . . . . . . 17 2.1.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.2 Growing and Shrinking . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 FastArrayStack: An Optimized ArrayStack . . . . . . . . . . . . . . . . . . 21 2.3 ArrayQueue: An Array-Based Queue . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 vii Contents Contents 2.4 ArrayDeque: Fast Deque Operations Using an Array . . . . . . . . . . . . . 25 2.4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 DualArrayDeque: Building a Deque from Two Stacks . . . . . . . . . . . . . 28 2.5.1 Balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.6 RootishArrayStack: A Space-Efficient Array Stack . . . . . . . . . . . . . 33 2.6.1 Analysis of Growing and Shrinking . . . . . . . . . . . . . . . . . . . 37 2.6.2 Space Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.6.4 Computing Square Roots . . . . . . . . . . . . . . . . . . . . . . . . 39 2.7 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Linked Lists 45 3.1 SLList: A Singly-Linked List . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.1.1 Queue Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 DLList: A Doubly-Linked List . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Adding and Removing . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 SEList: A Space-Efficient Linked List . . . . . . . . . . . . . . . . . . . . . 52 3.3.1 Space Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Finding Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.3 Adding an Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.4 Removing an Element . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.5 Amortized Analysis of Spreading and Gathering . . . . . . . . . . . 59 3.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4 Skiplists 63 4.1 The Basic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 SkiplistSSet: An Efficient SSet Implementation . . . . . . . . . . . . . . 65 4.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 viii Contents Contents 4.3 SkiplistList: An Efficient Random-Access List Implementation . . . . . 68 4.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 Analysis of Skiplists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.5 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5 Hash Tables 79 5.1 ChainedHashTable: Hashing with Chaining . . . . . . . . . . . . . . . . . . 79 5.1.1 Multiplicative Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 LinearHashTable: Linear Probing . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.1 Analysis of Linear Probing . . . . . . . . . . . . . . . . . . . . . . . 88 5.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.3 Tabulation Hashing . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.3 Hash Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.1 Hash Codes for Primitive Data Types . . . . . . . . . . . . . . . . . 92 5.3.2 Hash Codes for Compound Objects . . . . . . . . . . . . . . . . . . . 93 5.3.3 Hash Codes for Arrays and Strings . . . . . . . . . . . . . . . . . . . 94 5.4 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6 Binary Trees 101 6.1 BinaryTree: A Basic Binary Tree . . . . . . . . . . . . . . . . . . . . . . . 102 6.1.1 Recursive Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1.2 Traversing Binary Trees . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 BinarySearchTree: An Unbalanced Binary Search Tree . . . . . . . . . . . 106 6.2.1 Searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.2.2 Inserting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2.3 Deleting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.3 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 7 Random Binary Search Trees 115 7.1 Random Binary Search Trees . . . . . . . . . . . . . . . . . . . . . . . . . . 115 ix Contents Contents 7.1.1 Proof of Lemma 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.2 Treap: A Randomized Binary Search Tree . . . . . . . . . . . . . . . . . . . 120 7.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.3 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 8 Scapegoat Trees 131 8.1 ScapegoatTree: A Binary Search Tree with Partial Rebuilding . . . . . . . 132 8.1.1 Analysis of Correctness and Running-Time . . . . . . . . . . . . . . 135 8.1.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.2 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9 Red-Black Trees 141 9.1 2-4 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.1.1 Adding a Leaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.1.2 Removing a Leaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 9.2 RedBlackTree: A Simulated 2-4 Tree . . . . . . . . . . . . . . . . . . . . . . 144 9.2.1 Red-Black Trees and 2-4 Trees . . . . . . . . . . . . . . . . . . . . . 146 9.2.2 Left-Leaning Red-Black Trees . . . . . . . . . . . . . . . . . . . . . . 149 9.2.3 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.2.4 Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 9.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 9.4 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10 Heaps 163 10.1 BinaryHeap: An Implicit Binary Tree . . . . . . . . . . . . . . . . . . . . . 163 10.1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 10.2 MeldableHeap: A Randomized Meldable Heap . . . . . . . . . . . . . . . . 168 10.2.1 Analysis of merge(h1, h2) . . . . . . . . . . . . . . . . . . . . . . . . 170 10.2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.3 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 x Contents Contents 11 Sorting Algorithms 173 11.1 Comparison-Based Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 11.1.1 Merge-Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 11.1.2 Quicksort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 11.1.3 Heap-sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 11.1.4 A Lower-Bound for Comparison-Based Sorting . . . . . . . . . . . . 182 11.2 Counting Sort and Radix Sort . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11.2.1 Counting Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 11.2.2 Radix-Sort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 11.3 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 12 Graphs 191 12.1 AdjacencyMatrix: Representing a Graph by a Matrix . . . . . . . . . . . . 193 12.2 AdjacencyLists: A Graph as a Collection of Lists . . . . . . . . . . . . . . 195 12.3 Graph Traversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 12.3.1 Breadth-First Search . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 12.3.2 Depth-First Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 12.4 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 13 Data Structures for Integers 205 13.1 BinaryTrie: A digital search tree . . . . . . . . . . . . . . . . . . . . . . . 206 13.2 XFastTrie: Searching in Doubly-Logarithmic Time . . . . . . . . . . . . . . 211 13.3 YFastTrie: A Doubly-Logarithmic Time SSet . . . . . . . . . . . . . . . . 213 13.4 Discussion and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 xi Contents Contents xii Chapter 1 Introduction This chapter briefly reviews some of the main concepts used throughout the rest of the book. Section 1.1 describes the interfaces implemented by all the data structures described in this book. It should be considered required reading. The remaining sections discuss asymptotic (big-Oh) notation, probability and randomization, the model of computation, and the sample code and typesetting conventions. A reader with or without a background in these areas can easily skip them now and come back to them later if necessary. 1.1 Interfaces In discussing data structures, it is important to understand the difference between a data structure’s interface and its implementation. An interface describes what a data structure does, while an implementation describes how the data structure does it. An interface, sometimes also called an abstract data type, defines the set of operations supported by a data structure and the semantics, or meaning, of those operations. An interface tells us nothing about how the data structure implements these operations, it only provides the list of supported operations along with specifications about what types of arguments each operation accepts and the value returned by each operation. A data structure implementation on the other hand, includes the internal represen- tation of the data structure as well as the definitions of the algorithms that implement the operations supported by the data structure. Thus, there can be many implementations of a single interface. For example, in Chapter 2, we will see implementations of the List inter- face using arrays and in Chapter 3 we will see implementations of the List interface using pointer-based data structures. Each implements the same interface, List, but in different ways. 1 1. Introduction 1.1. Interfaces 1.1.1 The Queue, Stack, and Deque Interfaces The Queue interface represents a collection of elements to which we can add elements and remove the next element. More precisely, the operations supported by the Queue interface are • add(x): add the value x to the Queue • remove(): remove the next (previously added) value, y, from the Queue and return y Notice that the remove() operation takes no argument. The Queue’s queueing discipline decides which element should be removed. There are many possible queueing disciplines, the most common of which include FIFO, priority, and LIFO. A FIFO (first-in-first-out) Queue removes items in the same order they were added, much in the same way a queue (or line-up) works when checking out at a cash register in a grocery store. A priority Queue always removes the smallest element from the Queue, breaking ties arbitrarily. This is similar to the way many airlines manage upgrades to the business class on their flights. When a business-class seat becomes available it is given to the most important customer waiting on an upgrade. A very common queueing discipline is the LIFO (last-in-first-out) discipline. In a LIFO Queue, the most recently added element is the next one removed. This is best visualized in terms of a stack of plates; plates are placed on the top of the stack and also removed from the top of the stack. This structure is so common that it gets its own name: Stack. Often, when discussing a Stack, the names of add(x) and remove() are changed to push(x) and pop(); this is to avoid confusing the LIFO and FIFO queueing disciplines. A Deque is a generalization of both the FIFO Queue and LIFO Queue (Stack). A Deque represents a sequence of elements, with a front and a back. Elements can be added at the front of the sequence or the back of the sequence. The names of the operations on a Deque are self-explanatory: addFirst(x), removeFirst(), addLast(x), and removeLast(). Notice that a Stack can be implemented using only addFirst(x) and removeFirst() while a FIFO Queue can be implemented using only addLast(x) and removeFirst(). 1.1.2 The List Interface: Linear Sequences This book will talk very little about the FIFO Queue, Stack, or Deque interfaces. This is because these interfaces are subsumed by the List interface. A List represents a sequence, x0 , . . . , xn−1 , of values. The List interface includes the following operations: 2 1. Introduction 1.1. Interfaces 1. size(): return n, the length of the list 2. get(i): return the value xi 3. set(i, x): set the value of xi equal to x 4. add(i, x): add x at position i, displacing xi , . . . , xn−1 ; Set xj+1 = xj , for all j ∈ {n − 1, . . . , i}, increment n, and set xi = x 5. remove(i) remove the value xi , displacing xi+1 , . . . , xn−1 ; Set xj = xj+1 , for all j ∈ {i, . . . , n − 2} and decrement n Notice that these operations are easily sufficient to implement the Deque interface: addFirst(x) ⇒ add(0, x) removeFirst(x) ⇒ remove(0) addLast(x) ⇒ add(size(), x) removeLast(x) ⇒ remove(size() − 1) Although we will normally not discuss the Stack, Deque and FIFO Queue interfaces very often in subsequent chapters, the terms Stack and Deque are sometimes used in the names of data structures that implement the List interface. When this happens, it is to highlight the fact that these data structures can be used to implement the Stack or Deque interface very efficiently. For example, the ArrayDeque class is an implementation of the List interface that can implement the Deque operations in constant (amortized) time per operation. 1.1.3 The USet Interface: Unordered Sets The USet interface represents an unordered set of elements. This is a set in the mathematical sense. A USet contains n distinct elements; no element appears more than once; the elements are in no specific order. A USet supports the following operations: 1. size(): return the number, n, of elements in the set 2. add(x): add the element x to the set if not already present; Add x to the set provided that there is no element y in the set such that x equals y. Return true if x was added to the set and false otherwise. 3 1. Introduction 1.1. Interfaces 3. remove(x): remove x from the set; Find an element y in the set such that x equals y and remove y. Return y, or null if no such element exists. 4. find(x): find x in the set if it exists; Find an element y in the set such that y equals x. Return y, or null if no such element exists. These definitions are a bit fussy about distinguishing x, the element we are removing or finding, from y, the element we remove or find. This is because x and y might actually be distinct objects that are nevertheless treated as equal. This is a very useful distinction since it allows for the creation of dictionaries or maps that map keys onto values. This is done by creating a compound object called a Pair that contains a key and a value. Two Pairs are treated as equal if their keys are equal. By storing Pairs in a USet, we can find the value associated with any key k by creating a Pair, x, with key k and using the find(x) method. 1.1.4 The SSet Interface: Sorted Sets The SSet interface represents a sorted set of elements. An SSet stores elements from some total order, so that any two elements x and y can be compared. In code examples, this will be done with a method called compare(x, y) in which < 0 if x < y compare(x, y) > 0 if x > y = 0 if x = y An SSet supports the size(), add(x), and remove(x) methods with exactly the same se- mantics as in the USet interface. The difference between a USet and an SSet is in the find(x) method: 4. find(x): locate x in the sorted set; Find the smallest element y in the set such that y > x. Return y or null if no such element exists. This version of the find(x) operation is sometimes referred to as a successor search. It differs in a fundamental way from USet.find(x) since it returns a meaningful result even when there is no element in the set that is equal to x. 4 1. Introduction 1.2. Mathematical Background The distinction between the USet and SSet find(x) operations is very important and is very often missed. The extra functionality provided by an SSet usually comes with a price that includes both a larger running time and a higher implementation complexity. For example, the SSet implementations discussed in this book all have find(x) operations with running times that are at least logarithmic in the size of the set. On the other hand, the implementation of a USet as a ChainedHashTable in Chapter 5 has a find(x) operation that runs in constant expected time. When choosing which of these structures to use, one should always use a USet unless the extra functionality offered by an SSet is really needed. 1.2 Mathematical Background In this section, we review some mathematical notations and tools used throughout this book, including logarithms, big-Oh notation, and probability theory. 1.2.1 Logarithms In this book, the expression logb k denotes the base-b logarithm of k. That is, the unique value x that satisfies bx = k . Most of the logarithms in this book are base 2 (binary logarithms), in which case we drop the base, so that log k is shorthand for log2 k. Another logarithm that comes up several times in this book is the natural logarithm. Here we use the notation ln k to denote loge k, where e — Euler’s constant — is given by 1 n e = lim 1 + ≈ 2.71828 . n→∞ n The natural logarithm comes up frequently because it is the value of a particularly common integral: Z k 1/x dx = ln k . 1 Two of the most common manipulations we do with logarithms are removing them from an exponent: blogb k = k and changing the base of a logarithm: loga k logb k = . loga b 5 1. Introduction 1.2. Mathematical Background For example, we can use these two manipulations to compare the natural and binary loga- rithms log k log k ln k = = = (ln 2)(log k) ≈ 0.693147 log k . log e (ln e)/(ln 2) 1.2.2 Factorials In one or two places in this book, the factorial function is used. For a non-negative integer n, the notation n! (pronounced “n factorial”) denotes n! = 1 · 2 · 3 · · · · · n . Factorials appear because n! counts the number of distinct permutations, i.e., orderings, of n distinct elements. For the special case n = 0, 0! is defined as 1. The quantity n! can be approximated using Stirling’s Approximation: √ n n n! = 2πn eα(n) , e where 1 1 < α(n) < . 12n + 1 12n Stirling’s Approximation also approximates ln(n!): 1 ln(n!) = n ln n − n + ln(2πn) + α(n) 2 (In fact, Stirling’s Approximation is most easily proven by approximating ln(n!) = ln 1 + Rn ln 2 + · · · + ln n by the integral 1 ln n dn = n ln n − n + 1.) Related to the factorial function are the binomial coefficients. For a non-negative integer n and an integer k ∈ {0, . . . , n}, the notation nk denotes: n n! = . k k!(n − k)! n The binomial coefficient k (pronounced “n choose k”) counts the number of subsets of an n element set that have size k, i.e., the number of ways of choosing k distinct integers from the set {1, . . . , n}. 1.2.3 Asymptotic Notation When analyzing data structures in this book, we will want to talk about the running times of various operations. The exact running times will, of course, vary from computer to computer and even from run to run on an individual computer. Therefore, instead of 6 1. Introduction 1.2. Mathematical Background analyzing running times exactly, we will use the so-called big-Oh notation: For a function f (n), O(f (n)) denotes a set of functions, O(f (n)) = {g(n) : there exists c > 0, and n0 such that g(n) ≤ c · f (n) for all n ≥ n0 } . Thinking graphically, this set consists of the functions g(n) where c·f (n) starts to dominate g(n) when n is sufficiently large. We generally use asymptotic notation to simplify functions. For example, in place of 5n log n + 8n − 200 we can write, simply, O(n log n). This is proven as follows: 5n log n + 8n − 200 ≤ 5n log n + 8n ≤ 5n log n + 8n log n for n ≥ 2 (so that log n ≥ 1) ≤ 13n log n which demonstrates that the fundtion f (n) = 5n log n + 8n − 200 is in the set O(log n) using the constants c = 13 and n0 = 2. There are a number of useful shortcuts when using asymptotic notation. First: O(nc1 ) ⊂ O(nc2 ) , for any c1 < c2 . Second: For any constants a, b, c > 0, O(a) ⊂ O(log n) ⊂ O(nb ) ⊂ O(cn ) . These inclusion relations can be multiplied by any positive value, and they still hold. For example, multiplying by n yields: O(n) ⊂ O(n log n) ⊂ O(n1+b ) ⊂ O(ncn ) . Continuing in a long and distinguished tradition, we will abuse this notation by writing things like f1 (n) = O(f (n)) when what we really mean is f1 (n) ∈ O(f (n)). We will also make statements like “the running time of this operation is O(f (n))” when this statement should be “the running time of this operation is a member of O(f (n)).” These shortcuts are mainly to avoid awkward language and to make it easier to use asymptotic notation within strings of equations. A particularly strange example of this comes when we write statements like T (n) = 2 log n + O(1) . 7 1. Introduction 1.2. Mathematical Background Again, this would be more correctly written as T (n) ≤ 2 log n + [some member of O(1)] . The expression O(1) also brings up another issue. Since there is no variable in this expression, it may not be clear what variable is getting arbitrarily large. Without context, there is no way to tell. In the example above, since the only variable in the rest of the equation is n, we can assume that this should be read as T (n) = 2 log n + O(f (n)), where f (n) = 1. In a few cases, we will use asymptotic notation on functions with more than one variable. There seems to be no standard for this, but for our purposes, the following definition is sufficient: g(n1 , . . . , nk ) : there exists c > 0, and z such that O(f (n1 , . . . , nk )) = g(n1 , . . . , nk ) ≤ c · f (n1 , . . . , nk ) . for all n , . . . , n such that g(n , . . . , n ) ≥ z 1 k 1 k This definition captures the situation we really care about: when the arguments n1 , . . . , nk make g take on large values. This agrees with the univariate definition of O(f (n)) when f (n) is an increasing function of n. The reader should be warned that, although this works for our purposes, other texts may treat multivariate functions and asymptotic notation differently. 1.2.4 Randomization and Probability Some of the data structures presented in this book are randomized ; they make random choices that are independent of the data being stored in them or the operations being performed on them. For this reason, performing the same set of operations more than once using these structures could result in different running times. When analyzing these data structures we are interested in their average or expected running times. Formally, the running time of an operation on a randomized data structure is a random variable and we want to study its expected value. For a discrete random variable X taking on values in some countable universe U , the expected value of X, denoted by E[X] is given the by the formula X E[X] = x · Pr{X = x} . x∈U Here Pr{E} denotes the probability that the event E occurs. In all the examples in this book, these probabilities are only with respect to whatever random choices are made by the 8 1. Introduction 1.2. Mathematical Background randomized data structure; there is no assumption that the data stored in the structure is random or that the sequence of operations performed on the data structure is random. One of the most important properties of expected values is linearity of expectation: For any two random variables X and Y , E[X + Y ] = E[X] + E[Y ] . More generally, for any random variables X1 , . . . , Xk , " k # k X X E Xk = E[Xi ] . i=1 i=1 Linearity of expectation allows us to break down complicated random variables (like the left hand sides of the above equations) into sums of simpler random variables (the right hand sides). A useful trick, that we will use repeatedly, is that of defining indicator random variables. These binary variables are useful when we want to count something and are best illustrated by an example. Suppose we toss a fair coin k times and we want to know the expected number of times the coin comes up heads. Intuitively, we know the answer is k/2, but if we try to prove it using the definition of expected value, we get k X E[X] = i · Pr{X = i} i=0 Xk k = i· /2k i i=0 k−1 X k−1 =k· /2k i i=0 = k/2 . This requires that we know enough to calculate that Pr{X = i} = ki /2k , that we know the Pk binomial identity i ki = k k−1 i , and that we know the binomial identity k k i=0 i = 2 . Using indicator variables and linearity of expectation makes things much easier: For each i ∈ {1, . . . , k}, define the indicator random variable 1 if the ith coin toss is heads Ii = 0 otherwise. Then E[Ii ] = (1/2)1 + (1/2)0 = 1/2 . 9 1. Introduction 1.3. The Model of Computation Pk Now, X = i=1 Ii , so " k # X E[X] = E Ii i=1 k X = E[Ii ] i=1 Xk = 1/2 i=1 = k/2 . This is a bit more long-winded, but doesn’t require that we know any magical identities or compute any non-trivial probabilities. Even better: It agrees with the intuition that we expect half the coins to come up heads precisely because each individual coin has probability 1/2 of coming up heads. 1.3 The Model of Computation In this book, we will analyze the theoretical running times of operations on the data struc- tures we study. To do this precisely, we need a mathematical model of computation. For this, we use the w-bit word-RAM model. In this model, we have access to a random access memory consisting of cells, each of which stores a w-bit word. This implies a memory cell can represent, for example, any integer in the set {0, . . . , 2w − 1}. In the word-RAM model, basic operations on words take constant time. This in- cludes arithmetic operations (+, −, ∗, /, %), comparisons (<, >, =, ≤, ≥), and bitwise boolean operations (bitwise-AND, OR, and exclusive-OR). Any cell can be read or written in constant time. Our computer’s memory is man- aged by a memory management system from which we can allocate or deallocate a block of memory of any size we like. Allocating a block of memory of size k takes O(k) time and returns a reference to the newly-allocated memory block. This reference is small enough to be represented by a single word. The word-size, w, is a very important parameter of this model. The only assumption we will make on w is that it is at least w ≥ log n, where n is the number of elements stored in any of our data structures. This is a fairly modest assumption, since otherwise a word is not even big enough to count the number of elements stored in the data structure. Space is measured in words so that, when we talk about the amount of space used by a data structure, we are referring to the number of words of memory used by the structure. 10 1. Introduction 1.4. Code Samples All our data structures store values of a generic type T and we assume an element of type T occupies one word of memory. (In reality, we are storing references to objects of type T, and these references occupy only one word of memory.) The w-bit word-RAM model is a fairly close match for modern desktop computers when w = 32 or w = 64. The data structures presented in this book don’t use any special tricks that are not implementable on the JVM and most other architectures. 1.4 Code Samples The code samples in this book are written in the C++ programming language. However to make the book accessible even to readers not familiar with all of C++’s constructs and keywords, the code samples have been simplified. For example, a reader won’t find any of the keywords public, protected, private, or static. A reader also won’t find much discussion about class hierarchies. Which interfaces a particular class implements or which class it extends, if relevant to the discussion, will be clear from the accompanying text. These conventions should make most of the code samples understandable by anyone with a background in any of the languages from the ALGOL tradition, including B, C, C++, C#, D, Java, JavaScript, and so on. Readers who want the full details of all implementations are encouraged to look at the C++ source code that accompanies this book. This book mixes mathematical analysis of running times with C++ source code for the algorithms being analyzed. This means that some equations contain variables also found in the source code. These variables are typeset consistently, both within the source code and within equations. The most common such variable is the variable n that, without exception, always refers to the number of items currently stored in the data structure. 1.5 List of Data Structures The following table summarize the performance of data structures described in this book that implement each of the interfaces, List, USet, and SSet, described in Section 1.1. 11 1. Introduction 1.5. List of Data Structures List implementations get(i)/set(i, x) add(i, x)/remove(i) ArrayStack O(1) O(1 + n − i)A § 2.1 ArrayDeque O(1) O(1 + min{i, n − i})A § 2.4 DualArrayDeque O(1) O(1 + min{i, n − i})A § 2.5 RootishArrayStack O(1) O(1 + n − i)A § 2.6 DLList O(1 + min{i, n − i}) O(1 + min{i, n − i}) § 3.2 SEList O(1 + min{i, n − i}/b) O(b + min{i, n − i}/b)A § 3.3 SkiplistList O(log n)E O(log n)E § 4.3 USet implementations find(x) add(x)/remove(x) ChainedHashTable O(1)E O(1)AE § 5.1 LinearHashTable O(1)E O(1)AE § 5.2 SSet implementations find(x) add(x)/remove(x) SkiplistSSet O(log n)E O(log n)E § 4.2 Treap O(log n)E O(log n)E § 7.2 ScapegoatTree O(log n) O(log n)A § 8.1 RedBlackTree O(log n) O(log n) § 9.2 BinaryTrieI O(w) O(w) § 13.1 XFastTrieI O(log w)AE O(w)AE § 13.2 YFastTrieI O(log w)AE O(log w)AE § 13.3 (Priority) Queue implementations findMin() add(x)/remove() BinaryHeap O(1) O(log n)A § 10.1 MeldableHeap O(1) O(log n)E § 10.2 A Denotes an amortized running time. See Chapter 2. E Denotes an expected running time. See Section 1.2.4. F This structure can only store w-bit integer data. 12 1. Introduction 1.6. References 1.6 References The List, USet, and SSet interfaces described in Section 1.1 are influenced by the Java Collections Framework [45]. These are essentially simplified versions of the List, Set/Map, and SortedSet/SortedMap interfaces found in the Java Collections Framework. For more information on basic probability, especially as it relates to computer sci- ence, see the textbook by Ross [54]. Another good reference, that covers both asymptotic notation and probability, is the textbook by Graham, Knuth, and Patashnik [31]. 13 1. Introduction 1.6. References 14 Chapter 2 Array-Based Lists In this chapter, we study implementations of the List and Queue interfaces where the un- derlying data is stored in an array, called the backing array. The following table summarizes the running times of operations for the data structures presented in this chapter: get(i)/set(i, x) add(i, x)/remove(i) ArrayStack O(1) O(n − i) ArrayDeque O(1) O(min{i, n − i}) DualArrayDeque O(1) O(min{i, n − i}) RootishArrayStack O(1) O(n − i) Data structures that work by storing data in a single array have many advantages and limitations in common: • Arrays offer constant time access to any value in the array. This is what allows get(i) and set(i, x) to run in constant time. • Arrays are not very dynamic. Adding or removing an element near the middle of a list means that a large number of elements in the array need to be shifted to make room for the newly added element or to fill in the gap created by the deleted element. This is why the operations add(i, x) and remove(i) have running times that depend on n and i. • Arrays cannot expand or shrink. When the number of elements in the data structure exceeds the size of the backing array, a new array needs to be allocated and the data from the old array needs to be copied into the new array. This is an expensive operation. 15 2. Array-Based Lists The third point is important. The running times cited in the table above do not include the cost of growing and shrinking the backing array. We will see that, if carefully managed, the cost of growing and shrinking the backing array does not add much to the cost of an average operation. More precisely, if we start with an empty data structure, and perform any sequence of m add(i, x) or remove(i) operations, then the total cost of growing and shrinking the backing array, over the entire sequence of m operations is O(m). Although some individual operations are more expensive, the amortized cost, when amortized over all m operations, is only O(1) per operation. In this chapter, and throughout this book, it will be convenient to have arrays that keep track of their size. The usual C++ arrays do not do this, so we have defined a class, array, that keeps track of its length. The implementation of this class is straightforward. It is implemented as a standard C++ array, a, and an integer, length: array T *a; int length; The size of an array is specified at the time of creation: array array(int len) { length = len; a = new T[length]; } The elements of an array can be indexed: array T& operator[](int i) { assert(i >= 0 && i < length); return a[i]; } Finally, when one array is assigned to another, this is just a pointer manipulation that takes constant time: array array<T>& operator=(array<T> &b) { if (a != NULL) delete[] a; a = b.a; b.a = NULL; length = b.length; return *this; } 16 2. Array-Based Lists 2.1. ArrayStack: Fast Stack Operations Using an Array 2.1 ArrayStack: Fast Stack Operations Using an Array An ArrayStack implements the list interface using an array a, called the backing array. The list element with index i is stored in a[i]. At most times, a is larger than strictly necessary, so an integer n is used to keep track of the number of elements actually stored in a. In this way, the list elements are stored in a[0],. . . ,a[n − 1] and, at all times, a.length ≥ n. ArrayStack array<T> a; int n; int size() { return n; } 2.1.1 The Basics Accessing and modifying the elements of an ArrayStack using get(i) and set(i, x) is trivial. After performing any necessary bounds-checking we simply return or set, respectively, a[i]. ArrayStack T get(int i) { return a[i]; } T set(int i, T x) { T y = a[i]; a[i] = x; return y; } The operations of adding and removing elements from an ArrayStack are illustrated in Figure 2.1. To implement the add(i, x) operation, we first check if a is already full. If so, we call the method resize() to increase the size of a. How resize() is implemented will be discussed later. For now, it is sufficient to know that, after a call to resize(), we can be sure that a.length > n. With this out of the way, we now shift the elements a[i], . . . , a[n − 1] right by one position to make room for x, set a[i] equal to x, and increment n. ArrayStack void add(int i, T x) { if (n + 1 > a.length) resize(); for (int j = n; j > i; j--) a[j] = a[j - 1]; a[i] = x; n++; } 17 2. Array-Based Lists 2.1. ArrayStack: Fast Stack Operations Using an Array b r e d add(2,e) b r e e d add(5,r) b r e e d r add(5,e)∗ b r e e d r b r e e d e r remove(4) b r e e e r remove(4) b r e e r remove(4)∗ b r e e b r e e set(2,i) b r i e 0 1 2 3 4 5 6 7 8 9 10 11 Figure 2.1: A sequence of add(i, x) and remove(i) operations on an ArrayStack. Arrows denote elements being copied. Operations that result in a call to resize() are marked with an asterisk. 18 2. Array-Based Lists 2.1. ArrayStack: Fast Stack Operations Using an Array If we ignore the cost of the potential call to resize(), the cost of the add(i, x) operation is proportional to the number of elements we have to shift to make room for x. Therefore the cost of this operation (ignoring the cost of resizing a) is O(n − i + 1). Implementing the remove(i) operation is similar. We shift the elements a[i + 1], . . . , a[n − 1] left by one position (overwriting a[i]) and decrease the value of n. After doing this, we check if n is getting much smaller than a.length by checking if a.length ≥ 3n. If so, we call resize() to reduce the size of a. ArrayStack T remove(int i) { T x = a[i]; for (int j = i; j < n - 1; j++) a[j] = a[j + 1]; n--; if (a.length >= 3 * n) resize(); return x; } If we ignore the cost of the resize() method, the cost of a remove(i) operation is propor- tional to the number of elements we shift, which is O(n − i). 2.1.2 Growing and Shrinking The resize() method is fairly straightforward; it allocates a new array b whose size is 2n and copies the n elements of a into the first n positions in b, and then sets a to b. Thus, after a call to resize(), a.length = 2n. ArrayStack void resize() { array<T> b(max(2 * n, 1)); for (int i = 0; i < n; i++) b[i] = a[i]; a = b; } Analyzing the actual cost of the resize() operation is easy. It allocates an array b of size 2n and copies the n elements of a into b. This takes O(n) time. The running time analysis from the previous section ignored the cost of calls to resize(). In this section we analyze this cost using a technique known as amortized analysis. This technique does not try to determine the cost of resizing during each individual add(i, x) and remove(i) operation. Instead, it considers the cost of all calls to resize() during a sequence of m calls to add(i, x) or remove(i). In particular, we will show: 19 2. Array-Based Lists 2.1. ArrayStack: Fast Stack Operations Using an Array Lemma 2.1. If an empty ArrayList is created and any sequence of m ≥ 1 calls to add(i, x) and remove(i) are performed, then the total time spent during all calls to resize() is O(m). Proof. We will show that anytime resize() is called, the number of calls to add or remove since the last call to resize() is at least n/2 − 1. Therefore, if ni denotes the value of n during the ith call to resize() and r denotes the number of calls to resize(), then the total number of calls to add(i, x) or remove(i) is at least r X (ni /2 − 1) ≤ m , i=1 which is equivalent to r X ni ≤ 2m + 2r . i=1 On the other hand, the total time spent during all calls to resize() is r X O(ni ) ≤ O(m + r) = O(m) , i=1 which will prove the lemma since r is not more than m. All that remains is to show that the number of calls to add(i, x) or remove(i) between the (i − 1)th and the ith call to resize() is at least ni /2. There are two cases to consider. In the first case, resize() is being called by add(i, x) because the backing array a is full, i.e., a.length = n = ni . Consider the previous call to resize(): After this previous call, the size of a was a.length, but the number of elements stored in a was at most a.length/2 = ni /2. But now the number of elements stored in a is ni = a.length, so there must have been at least ni /2 calls to add(i, x) since the previous call to resize(). The second case to consider is when resize() is being called by remove(i) because a.length ≥ 3n = 3ni . Again, after the previous call to resize() the number of elements stored in a was at least a.length/2 − 1.1 Now there are ni ≤ a.length/3 elements stored in a. Therefore, the number of remove(i) operations since the last call to resize() is at least a.length/2 − 1 − a.length/3 = a.length/6 − 1 = (a.length/3)/2 − 1 ≥ ni /2 − 1 . In either case, the number of calls to add(i, x) or remove(i) that occur between the (i−1)th call to resize() and the ith call to resize() is at least ni /2 − 1, as required to complete the proof. 1 The − 1 in this formula accounts for the special case that occurs when n = 0 and a.length = 1. 20 2. Array-Based Lists 2.2. FastArrayStack: An Optimized ArrayStack 2.1.3 Summary The following theorem summarizes the performance of an ArrayStack: Theorem 2.1. An ArrayStack implements the List interface. Ignoring the cost of calls to resize(), an ArrayStack supports the operations • get(i) and set(i, x) in O(1) time per operation; and • add(i, x) and remove(i) in O(1 + n − i) time per operation. Furthermore, beginning with an empty ArrayStack, any sequence of m add(i, x) and remove(i) operations results in a total of O(m) time spent during all calls to resize(). The ArrayStack is an efficient way to implement a Stack. In particular, we can implement push(x) as add(n, x) and pop() as remove(n − 1), in which case these operations will run in O(1) amortized time. 2.2 FastArrayStack: An Optimized ArrayStack Much of the work done by an ArrayStack involves shifting (by add(i, x) and remove(i)) and copying (by resize()) of data. In the implementations shown above, this was done using for loops. It turns out that many programming environments have specific functions that are very efficient at copying and moving blocks of data. In the C there are the memcpy(d, s, n) and memmove(d, s, n) functions. In C++ there is the std :: copy(a0, a1, b) and algorithm. In Java there is the System.arraycopy(s, i, d, j, n) method. FastArrayStack void resize() { array<T> b(max(1, 2*n)); std::copy(a+0, a+n, b+0); a = b; } void add(int i, T x) { if (n + 1 > a.length) resize(); std::copy_backward(a+i, a+n, a+n); a[i] = x; n++; } These functions are usually highly optimized and may even use special machine in- structions that can do this copying much faster than we could do using a for loop. Although 21 2. Array-Based Lists 2.3. ArrayQueue: An Array-Based Queue using these functions does not asymptotically decrease the running times, it can still be a worthwhile optimization. In the C++ implementations here, the use of std :: copy(a0, a1, b) resulted in speedups of a factor of 2-3 depending on the types of operations performed. Your mileage may vary. 2.3 ArrayQueue: An Array-Based Queue In this section, we present the ArrayQueue data structure, which implements a FIFO (first- in-first-out) queue; elements are removed (using the remove() operation) from the queue in the same order they are added (using the add(x) operation). Notice that an ArrayStack is a poor choice for an implementation of a FIFO queue. The reason is that we must choose one end of the list to add to and then remove from the other end. One of the two operations must work on the head of the list, which involves calling add(i, x) or remove(i) with a value of i = 0. This gives a running time of Θ(n). To obtain an efficient array-based implementation of a queue, we first notice that the problem would be easy if we had an infinite array a. We could maintain one index j that keeps track of the next element to remove and an integer n that counts the number of elements in the queue. The queue elements would always be stored in a[j], a[j + 1], . . . , a[j + n − 1] . Initially, both j and n would be set to 0. To add an element, we would place it in a[j + n] and increment n. To remove an element, we would remove it from a[j], increment j, and decrement n. Of course, the problem with this solution is that it requires an infinite array. An ArrayQueue simulates this by using a finite array a and modular arithmetic. This is the kind of arithmetic used when we are talking about the time of day. For example 10 o’clock plus 5 hours gives 3 o’clock. Formally, we say that 10 + 5 = 15 ≡ 3 (mod 12) . We read the latter part of this equation as “15 is congruent to 3 modulo 12.” We can also treat mod as a binary operator, so that 15 mod 12 = 3 . More generally, for an integer a and positive integer m, a mod m is the unique integer r ∈ {0, . . . , m − 1} such that a = r + km for some integer k. Less formally, the 22 2. Array-Based Lists 2.3. ArrayQueue: An Array-Based Queue value r is the remainder we get when we divide a by m. In many programming languages, including C++, the mod operator is represented using the % symbol.2 Modular arithmetic is useful for simulating an infinite array, since i mod a.length always gives a value in the range 0, . . . , a.length − 1. Using modular arithmetic we can store the queue elements at array locations a[j%a.length], a[(j + 1)%a.length], . . . , a[(j + n − 1)%a.length] . This treats a like a circular array in which array indices exceeding a.length − 1 “wrap around” to the beginning of the array. The only remaining thing to worry about is taking care that the number of elements in the ArrayQueue does not exceed the size of a. ArrayQueue array<T> a; int j; int n; A sequence of add(x) and remove() operations on an ArrayQueue is illustrated in Figure 2.2. To implement add(x), we first check if a is full and, if necessary, call resize() to increase the size of a. Next, we store x in a[(j + n)%a.length] and increment n. ArrayQueue bool add(T x) { if (n + 1 > a.length) resize(); a[(j+n) % a.length] = x; n++; return true; } To implement remove() we first store a[j] so that we can return it later. Next, we decrement n and increment j (modulo a.length) by setting j = (j + 1) mod a.length. Finally, we return the stored value of a[j]. If necessary, we may call resize() to decrease the size of a. ArrayQueue T remove() { T x = a[j]; j = (j + 1) % a.length; 2 This is sometimes referred to as the brain-dead mod operator since it does not correctly implement the mathematical mod operator when the first argument is negative. 23 2. Array-Based Lists 2.3. ArrayQueue: An Array-Based Queue j = 2, n = 3 a b c add(d) j = 2, n = 4 a b c d add(e) j = 2, n = 5 e a b c d remove() j = 3, n = 4 e b c d add(f) j = 3, n = 5 e f b c d add(g) j = 3, n = 6 e f g b c d add(h)∗ j = 0, n = 6 b c d e f g j = 0, n = 7 b c d e f g h remove() j = 1, n = 6 c d e f g h 0 1 2 3 4 5 6 7 8 9 10 11 Figure 2.2: A sequence of add(x) and remove(i) operations on an ArrayQueue. Arrows denote elements being copied. Operations that result in a call to resize() are marked with an asterisk. 24 2. Array-Based Lists 2.4. ArrayDeque: Fast Deque Operations Using an Array n--; if (a.length >= 3*n) resize(); return x; } Finally, the resize() operation is very similar to the resize() operation of ArrayStack. It allocates a new array b of size 2n and copies a[j], a[(j + 1)%a.length], . . . , a[(j + n − 1)%a.length] onto b[0], b[1], . . . , b[n − 1] and sets j = 0. ArrayQueue void resize() { array<T> b(max(1, 2*n)); for (int k = 0; k < n; k++) b[k] = a[(j+k)%a.length]; a = b; } 2.3.1 Summary The following theorem summarizes the performance of the ArrayQueue data structure: Theorem 2.2. An ArrayQueue implements the (FIFO) Queue interface. Ignoring the cost of calls to resize(), an ArrayQueue supports the operations add(x) and remove() in O(1) time per operation. Furthermore, beginning with an empty ArrayQueue, any sequence of m add(i, x) and remove(i) operations results in a total of O(m) time spent during all calls to resize(). 2.4 ArrayDeque: Fast Deque Operations Using an Array The ArrayQueue from the previous section is a data structure for representing a sequence that allows us to efficiently add to one end of the sequence and remove from the other end. The ArrayDeque data structure allows for efficient addition and removal at both ends. This structure implements the List interface using the same circular array technique used to represent an ArrayQueue. 25 2. Array-Based Lists 2.4. ArrayDeque: Fast Deque Operations Using an Array j = 0, n = 8 a b c d e f g h remove(2) j = 1, n = 7 a b d e f g h add(4,x) j = 1, n = 8 a b d e x f g h add(3,y) j = 0, n = 9 a b d y e x f g h add(4,z) j = 11, n = 10 b d y z e x f g h a 0 1 2 3 4 5 6 7 8 9 10 11 Figure 2.3: A sequence of add(i, x) and remove(i) operations on an ArrayDeque. Arrows denote elements being copied. ArrayDeque array<T> a; int j; int n; The get(i) and set(i, x) operations on an ArrayDeque are straightforward. They get or set the array element a[(j + i) mod a.length]. ArrayDeque T get(int i) { return a[(j + i) % a.length]; } T set(int i, T x) { T y = a[(j + i) % a.length]; a[(j + i) % a.length] = x; return y; } The implementation of add(i, x) is a little more interesting. As usual, we first check if a is full and, if necessary, call resize() to resize a. Remember that we want this operation to be fast when i is small (close to 0) or when i is large (close to n). Therefore, we check if i < n/2. If so, we shift the elements a[0], . . . , a[i − 1] left by one position. Otherwise (i ≥ n/2), we shift the elements a[i], . . . , a[n − 1] right by one position. See Figure 2.3 for an illustration of add(i, x) and remove(x) operations on an ArrayDeque. 26 2. Array-Based Lists 2.4. ArrayDeque: Fast Deque Operations Using an Array ArrayDeque void add(int i, T x) { if (n + 1 > a.length) resize(); if (i < n/2) { // shift a[0],..,a[i-1] left one position j = (j == 0) ? a.length - 1 : j - 1; for (int k = 0; k <= i-1; k++) a[(j+k)%a.length] = a[(j+k+1)%a.length]; } else { // shift a[i],..,a[n-1] right one position for (int k = n; k > i; k--) a[(j+k)%a.length] = a[(j+k-1)%a.length]; } a[(j+i)%a.length] = x; n++; } By doing the shifting in this way, we guarantee that add(i, x) never has to shift more than min{i, n − i} elements. Thus, the running time of the add(i, x) operation (ignoring the cost of a resize() operation) is O(1 + min{i, n − i}). The remove(i) operation is similar. It either shifts elements a[0], . . . , a[i − 1] right by one position or shifts the elements a[i + 1], . . . , a[n − 1] left by one position depending on whether i < n/2. Again, this means that remove(i) never spends more than O(1 + min{i, n − i}) time to shift elements. ArrayDeque T remove(int i) { T x = a[(j+i)%a.length]; if (i < n/2) { // shift a[0],..,[i-1] right one position for (int k = i; k > 0; k--) a[(j+k)%a.length] = a[(j+k-1)%a.length]; j = (j + 1) % a.length; } else { // shift a[i+1],..,a[n-1] left one position for (int k = i; k < n-1; k++) a[(j+k)%a.length] = a[(j+k+1)%a.length]; } n--; if (3*n < a.length) resize(); return x; } 2.4.1 Summary The following theorem summarizes the performance of the ArrayDeque data structure: 27 2. Array-Based Lists 2.5. DualArrayDeque: Building a Deque from Two Stacks Theorem 2.3. An ArrayDeque implements the List interface. Ignoring the cost of calls to resize(), an ArrayDeque supports the operations • get(i) and set(i, x) in O(1) time per operation; and • add(i, x) and remove(i) in O(1 + min{i, n − i}) time per operation. Furthermore, beginning with an empty ArrayDeque, any sequence of m add(i, x) and remove(i) operations results in a total of O(m) time spent during all calls to resize(). 2.5 DualArrayDeque: Building a Deque from Two Stacks Next, we present another data structure, the DualArrayDeque that achieves the same per- formance bounds as an ArrayDeque by using two ArrayStacks. Although the asymptotic performance of the DualArrayDeque is no better than that of the ArrayDeque, it is still worth studying since it offers a good example of how to make a sophisticated data structure by combining two simpler data structures. A DualArrayDeque represents a list using two ArrayStacks. Recall that an ArrayStack is fast when the operations on it modify elements near the end. A DualArrayDeque places two ArrayStacks, called front and back, back-to-back so that operations are fast at either end. DualArrayDeque ArrayStack<T> front; ArrayStack<T> back; A DualArrayDeque does not explicitly store the number, n, of elements it contains. It doesn’t need to, since it contains n = front.size()+back.size() elements. Nevertheless, when analyzing the DualArrayDeque we will still use n to denote the number of elements it contains. DualArrayDeque int size() { return front.size() + back.size(); } The front ArrayStack contains list elements with indices 0, . . . , front.size() − 1, but stores them in reverse order. The back ArrayStack contains list elements with indices front.size(), . . . , size()−1 in the normal order. In this way, get(i) and set(i, x) translate into appropriate calls to get(i) or set(i, x) on either front or back, which take O(1) time per operation. 28 2. Array-Based Lists 2.5. DualArrayDeque: Building a Deque from Two Stacks front back a b c d add(3,x) a b c x d add(4,y) a b c x y d remove(0)∗ b c x y d b c x y d 4 3 2 1 0 0 1 2 3 4 Figure 2.4: A sequence of add(i, x) and remove(i) operations on a DualArrayDeque. Arrows denote elements being copied. Operations that result in a rebalancing by balance() are marked with an asterisk. DualArrayDeque T get(int i) { if (i < front.size()) { return front.get(front.size() - i - 1); } else { return back.get(i - front.size()); } } T set(int i, T x) { if (i < front.size()) { return front.set(front.size() - i - 1, x); } else { return back.set(i - front.size(), x); } } Note that, if an index i < front.size(), then it corresponds to the element of front at position front.size() − i − 1, since the elements of front are stored in reverse order. Adding and removing elements from a DualArrayDeque is illustrated in Figure 2.4. The add(i, x) operation manipulates either front or back, as appropriate: DualArrayDeque void add(int i, T x) { 29 2. Array-Based Lists 2.5. DualArrayDeque: Building a Deque from Two Stacks if (i < front.size()) { front.add(front.size() - i, x); } else { back.add(i - front.size(), x); } balance(); } The add(i, x) method performs rebalancing of the two ArrayStacks front and back, by calling the balance() method. The implementation of balance() is described below, but for now it is sufficient to know that balance() ensures that, unless size() < 2, front.size() and back.size() do not differ by more than a factor of 3. In particular, 3 · front.size() ≥ back.size() and 3 · back.size() ≥ front.size(). Next we analyze the cost of add(i, x), ignoring the cost of the balance() operation. If i < front.size(), then add(i, x) becomes front.add(front.size() − i − 1, x). Since front is an ArrayStack, the cost of this is O(front.size() − (front.size() − i − 1) + 1) = O(i + 1) . (2.1) On the other hand, if i ≥ front.size(), then add(i, x) becomes back.add(i − front.size(), x). The cost of this is O(back.size() − (i − front.size()) + 1) = O(n − i + 1) . (2.2) Notice that the first case (2.1) occurs when i < n/4. The second case (2.2) occurs when i ≥ 3n/4. When n/4 ≤ i < 3n/4, we can’t be sure whether the operation affects front or back, but in either case, the operation takes O(n) = O(i) = O(n − i) time, since i ≥ n/4 and n − i > n/4. Summarizing the situation, we have O(1 + i) if i < n/4 Running time of add(i, x) ≤ O(n) if n/4 ≤ i < 3n/4 O(1 + n − i) if i ≥ 3n/4 Thus, the running time of add(i, x) (ignoring the cost of the call to balance()) is O(1 + min{i, n − i}). The remove(i) operation, and its analysis, is similar to the add(i, x) operation. DualArrayDeque T remove(int i) { T x; 30 2. Array-Based Lists 2.5. DualArrayDeque: Building a Deque from Two Stacks if (i < front.size()) { x = front.remove(front.size()-i-1); } else { x = back.remove(i-front.size()); } balance(); return x; } 2.5.1 Balancing Finally, we study the balance() operation performed by add(i, x) and remove(i). This operation is used to ensure that neither front nor back gets too big (or too small). It ensures that, unless there are fewer than 2 elements, each of front and back contain at least n/4 elements. If this is not the case, then it moves elements between them so that front and back contain exactly bn/2c elements and dn/2e elements, respectively. DualArrayDeque void balance() { if (3*front.size() < back.size() || 3*back.size() < front.size()) { int n = front.size() + back.size(); int nf = n/2; array<T> af(max(2*nf, 1)); for (int i = 0; i < nf; i++) { af[nf-i-1] = get(i); } int nb = n - nf; array<T> ab(max(2*nb, 1)); for (int i = 0; i < nb; i++) { ab[i] = get(nf+i); } front.a = af; front.n = nf; back.a = ab; back.n = nb; } } There is not much to analyze. If the balance() operation does rebalancing, then it moves Θ(n) elements and this takes O(n) time. This is bad, since balance() is called with each call to add(i, x) and remove(i). However, the following lemma shows that, on average, balance() only spends a constant amount of time per operation. 31 2. Array-Based Lists 2.5. DualArrayDeque: Building a Deque from Two Stacks Lemma 2.2. If an empty DualArrayDeque is created and any sequence of m ≥ 1 calls to add(i, x) and remove(i) are performed, then the total time spent during all calls to balance() is O(m). Proof. We will show that, if balance() is forced to shift elements, then the number of add(i, x) and remove(i) operations since the last time balance() shifted any elements is at least n/2 − 1. As in the proof of Lemma 2.1, this is sufficient to prove that the total time spent by balance() is O(m). We will perform our analysis using the potential method. Define the potential of the DualArrayDeque as Φ = |front.size() − back.size()| . The interesting thing about this potential is that a call to add(i, x) or remove(i) that does not do any balancing can increase the potential by at most 1. Observe that, immediately after a call to balance() that shifts elements, the poten- tial, Φ0 , is at most 1, since Φ0 = |bn/2c − dn/2e| ≤ 1 . Consider the situation immediately before a call to balance() that shifts elements and suppose, without loss of generality, that balance() is shifting elements because 3front.size() < back.size(). Notice that, in this case, n = front.size() + back.size() < back.size()/3 + back.size() 4 = back.size() 3 Furthermore, the potential at this point in time is Φ1 = back.size() − front.size() > back.size() − back.size()/3 2 = back.size() 3 2 3 > × n 3 4 = n/2 Therefore, the number of calls to add(i, x) or remove(i) since the last time balance() shifted elements is at least Φ1 − Φ0 > n/2 − 1. This completes the proof. 32 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack 2.5.2 Summary The following theorem summarizes the performance of a DualArrayStack Theorem 2.4. A DualArrayDeque implements the List interface. Ignoring the cost of calls to resize() and balance(), a DualArrayDeque supports the operations • get(i) and set(i, x) in O(1) time per operation; and • add(i, x) and remove(i) in O(1 + min{i, n − i}) time per operation. Furthermore, beginning with an empty DualArrayDeque, any sequence of m add(i, x) and remove(i) operations results in a total of O(m) time spent during all calls to resize() and balance(). 2.6 RootishArrayStack: A Space-Efficient Array Stack One of the drawbacks of all previous data structures in this chapter is that, because they store their data in one or two arrays, and they avoid resizing these arrays too often, the arrays are frequently not very full. For example, immediately after a resize() operation on an ArrayStack, the backing array a is only half full. Even worse, there are times when only 1/3 of a contains data. In this section, we discuss a data structure, the RootishArrayStack, that addresses √ the problem of wasted space. The RootishArrayStack stores n elements using O( n) √ arrays. In these arrays, at most O( n) array locations are unused at any time. All remaining √ array locations are used to store data. Therefore, these data structures waste at most O( n) space when storing n elements. A RootishArrayStack stores its elements in a list of r arrays called blocks that are numbered 0, 1, . . . , r − 1. See Figure 2.5. Block b contains b + 1 elements. Therefore, all r blocks contain a total of 1 + 2 + 3 + · · · + r = r(r + 1)/2 elements. The above formula (allegedly discovered by the mathematician Gauß at the age of 9) can be obtained as shown in Figure 2.6. RootishArrayStack ArrayStack<T*> blocks; int n; The elements of the list are laid out in the blocks as we might expect. The list element with index 0 is stored in block 0, the elements with list indices 1 and 2 are stored in 33 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack blocks a b c d e f g h add(2,x) a b x c d e f g h remove(1) a x c d e f g h remove(7) a x c d e f g remove(6) a x c d e f 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 2.5: A sequence of add(i, x) and remove(i) operations on a RootishArrayStack. Arrows denote elements being copied. ... .. . r .. . .. . ... r+1 Figure 2.6: The number of white squares is 1 + 2 + 3 + · · · + r. The number of shaded squares is the same. Together the white and shaded squares make a rectangle consisting of r(r + 1) squares. 34 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack block 1, the elements with list indices 3, 4, and 5 are stored in block 2, and so on. The main problem we have to address is that of determining, given an index i, which block contains i as well as the index corresponding to i within that block. Determining the index of i within its block turns out to be easy. If index i is in block b, then the number of elements in blocks 0, . . . , b − 1 is b(b + 1)/2. Therefore, i is stored at location j = i − b(b + 1)/2 within block b. Somewhat more challenging is the problem of determining the value of b. The number of elements that have indices less than or equal to i is i + 1. On the other hand, the number of elements in blocks 0,. . . ,b is (b + 1)(b + 2)/2. Therefore, b is the smallest integer such that (b + 1)(b + 2)/2 ≥ i + 1 . We can rewrite this equation as b2 + 3b − 2i ≥ 0 . The corresponding quadratic equation b2 + 3b − 2i = 0 has two solutions: b = (−3 + √ √ 9 + 8i)/2 and b = (−3 − 9 + 8i)/2. The second solution makes no sense in our ap- plication since it always gives a negative value. Therefore, we obtain the solution b = √ (−3 + 9 + 8i)/2. In general, this solution is not an integer, but going back to our inequal- √ ity, we want the smallest integer b such that b ≥ (−3 + 9 + 8i)/2. This is simply l √ m b = (−3 + 9 + 8i)/2 . RootishArrayStack int i2b(int i) { double db = (-3.0 + sqrt(9 + 8*i)) / 2.0; int b = (int)ceil(db); return b; } With this out of the way, the get(i) and set(i, x) methods are straightforward. We first compute the appropriate block b and the appropriate index j within the block and then perform the appropriate operation: RootishArrayStack T get(int i) { int b = i2b(i); int j = i - b*(b+1)/2; 35 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack return blocks.get(b)[j]; } T set(int i, T x) { int b = i2b(i); int j = i - b*(b+1)/2; T y = blocks.get(b)[j]; blocks.get(b)[j] = x; return y; } If we use any of the data structures in this chapter for representing the blocks list, then get(i) and set(i, x) will each run in constant time. The add(i, x) method will, by now, look familiar. We first check if our data structure is full, by checking if the number of blocks r is such that r(r + 1)/2 = n and, if so, we call grow() to add another block. With this done, we shift elements with indices i, . . . , n − 1 to the right by one position to make room for the new element with index i: RootishArrayStack void add(int i, T x) { int r = blocks.size(); if (r*(r+1)/2 < n + 1) grow(); n++; for (int j = n-1; j > i; j--) set(j, get(j-1)); set(i, x); } The grow() method does what we expect. It adds a new block: RootishArrayStack void grow() { blocks.add(blocks.size(), new T[blocks.size()+1]); } Ignoring the cost of the grow() operation, the cost of an add(i, x) operation is dominated by the cost of shifting and is therefore O(1 + n − i), just like an ArrayStack. The remove(i) operation is similar to add(i, x). It shifts the elements with indices i + 1, . . . , n left by one position and then, if there is more than one empty block, it calls the shrink() method to remove all but one of the unused blocks: RootishArrayStack T remove(int i) { T x = get(i); 36 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack for (int j = i; j < n-1; j++) set(j, get(j+1)); n--; int r = blocks.size(); if ((r-2)*(r-1)/2 >= n) shrink(); return x; } RootishArrayStack void shrink() { int r = blocks.size(); while (r > 0 && (r-2)*(r-1)/2 >= n) { delete [] blocks.remove(blocks.size()-1); r--; } } Once again, ignoring the cost of the shrink() operation, the cost of a remove(i) operation is dominated by the cost of shifting and is therefore O(n − i). 2.6.1 Analysis of Growing and Shrinking The above analysis of add(i, x) and remove(i) does not account for the cost of grow() and shrink(). Note that, unlike the ArrayStack.resize() operation, grow() and shrink() do not do any copying of data. They only allocate or free an array of size r. In some environments, this takes only constant time, while in others, it may require Θ(r)) time. We note that, immediately after a call to grow() or shrink(), the situation is clear. The final block is completely empty and all other blocks are completely full. Another call to grow() or shrink() will not happen until at least r−1 elements have been added or removed. Therefore, even if grow() and shrink() take O(r) time, this cost can be amortized over at least r − 1 add(i, x) or remove(i) operations, so that the amortized cost of grow() and shrink() is O(1) per operation. 2.6.2 Space Usage Next, we analyze the amount of extra space used by a RootishArrayStack. In particular, we want to count any space used by a RootishArrayStack that is not an array element currently used to hold a list element. We call all such space wasted space. The remove(i) operation ensures that a RootishArrayStack never has more than 2 blocks that are not completely full. The number of blocks, r, used by a RootishArrayStack 37 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack that stores n elements therefore satisfies (r − 2)(r − 1) ≤ n Again, using the quadratic equation on this gives √ √ r ≤ (3 + 1 + 4n)/2 = O( n) The last two blocks have sizes r and r − 1, so the space wasted by these two blocks is at √ most 2r − 1 = O( n). If we store the blocks in (for example) an ArrayList, then the √ amount of space wasted by the List that stores those r blocks is also O(r) = O( n). The other space needed for storing n and other accounting information is O(1). Therefore, the √ total amount of wasted space in a RootishArrayStack is O( n). Next, we argue that this space usage is optimal for any data structure that starts out empty and can support the addition of one item at a time. More precisely, we will show that, at some point during the addition of n items, the data structure is wasting an amount √ of space at least in n (though it may be only wasted for a moment). Suppose we start with an empty data structure and we add n items one at a time. At the end of this process, all n items are stored in the structure and they are distributed √ among a collection of r memory blocks. If r ≥ n, then the data structure must be using r pointers (or references) to keep track of these r blocks, and this is wasted space. On the √ other hand, if r < n then, by the pigeonhole principle, some block must have size at least √ n/r > n. Consider the moment at which this block was first allocated. Immediately after √ it was allocated, this block was empty, and was therefore wasting n space. Therefore, at √ some point in time during the insertion of n elements, the data structure was wasting n space. 2.6.3 Summary The following theorem summarizes the performance of the RootishArrayStack data struc- ture: Theorem 2.5. A RootishArrayStack implements the List interface. Ignoring the cost of calls to grow() and shrink(), a RootishArrayStack supports the operations • get(i) and set(i, x) in O(1) time per operation; and • add(i, x) and remove(i) in O(1 + n − i) time per operation. 38 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack Furthermore, beginning with an empty RootishArrayStack, any sequence of m add(i, x) and remove(i) operations results in a total of O(m) time spent during all calls to grow() and shrink(). The space (measured in words)3 used by a RootishArrayStack that stores n elements √ is n + O( n). 2.6.4 Computing Square Roots A reader who has had some exposure to models of computation may notice that the RootishArrayStack, as described above, does not fit into the usual word-RAM model of computation (Section 1.3) because it requires taking square roots. The square root oper- ation is generally not considered a basic operation and is therefore not usually part of the word-RAM model. In this section, we take time to show that the square root operation can be imple- √ mented efficiently. In particular, we show that for any integer x ∈ {0, . . . , n}, b xc can √ be computed in constant-time, after O( n) preprocessing that creates two arrays of length √ O( n). The following lemma shows that we can reduce the problem of computing the square root of x to the square root of a related value x0 . √ √ √ Lemma 2.3. Let x ≥ 1 and let x0 = x − a, where 0 ≤ a ≤ x. Then x0 ≥ x − 1. Proof. It suffices to show that q √ √ x− x≥ x−1 . Square both sides of this inequality to get √ √ x− x≥x−2 x+1 and gather terms to get √ x≥1 which is clearly true for any x ≥ 1. Start by restricting the problem a little, and assume that 2r ≤ x < 2r+1 , so that blog xc = r, i.e., x is an integer having r + 1 bits in its binary representation. We can take √ √ x0 = x − (x mod 2br/2c ). Now, x0 satisfies the conditions of Lemma 2.3, so x − x0 ≤ 1. Furthermore, x0 has all of its lower-order br/2c bits equal to 0, so there are only √ 2r+1−br/2c ≤ 4 · 2r/2 ≤ 4 x 3 Recall Section 1.3 for a discussion of how memory is measured. 39 2. Array-Based Lists 2.6. RootishArrayStack: A Space-Efficient Array Stack possible values of x0 . This means that we can use an array, sqrttab, that stores the value √ of b x0 c for each possible value of x0 . A little more precisely, we have jp k sqrttab[i] = i2br/2c . √ In this way, sqrttab[i] is within 2 ofx for all x ∈ {i2br/2c , . . . , (i + 1)2br/2c − 1}. Stated √ √ another way, the array entry s = sqrttab[x>>br/2c] is either equal to b xc, b xc − 1, or √ √ b xc − 2. From s we can determine the value of b xc by incrementing s until (s + 1)2 > x. FastSqrt int sqrt(int x, int r) { int s = sqrtab[x>>r/2]; while ((s+1)*(s+1) <= x) s++; // executes at most twice return s; } Now, this only works for x ∈ {2r , . . . , 2r+1 − 1} and sqrttab is a special table that only works for a particular value of r = blog xc. To overcome this, we could compute blog nc different sqrttab arrays, one for each possible value of blog xc. The sizes of these tables √ form an exponential sequence whose largest value is at most 4 n, so the total size of all √ tables is O( n). However, it turns out that more than one sqrttab array is unnecessary; we only need one sqrttab array for the value r = blog nc. Any value x with log x = r0 < r can be 0 upgraded by multiplying x by 2r−r and using the equation √ 0 √ 2r−r0 x = 2(r−r )/2 x . 0 The quantity 2r−r x is in the range {2r , . . . , 2r+1 − 1} so we can look up its square root in √ sqrttab. The following code implements this idea to compute b xc for all non-negative integers x in the range {0, . . . , 230 − 1} using an array, sqrttab, of size 216 . FastSqrt int sqrt(int x) { int rp = log(x); int upgrade = ((r-rp)/2) * 2; int xp = x << upgrade; // xp has r or r-1 bits int s = sqrtab[xp>>(r/2)] >> (upgrade/2); while ((s+1)*(s+1) <= x) s++; // executes at most twice return s; } Something we have taken for granted thus far is the question of how to compute r0 = blog xc. Again, this is a problem that can be solved with an array, logtab, of size 40 2. Array-Based Lists 2.7. Discussion and Exercises 2r/2 . In this case, the code is particularly simple, since blog xc is just the index of the most significant 1 bit in the binary representation of x. This means that, for x > 2r/2 , we can right-shift the bits of x by r/2 positions before using it as an index into logtab. The following code does this using an array logtab of size 216 to compute blog xc for all x in the range {1, . . . , 232 − 1} FastSqrt int log(int x) { if (x >= halfint) return 16 + logtab[x>>16]; return logtab[x]; } Finally, for completeness, we include the following code that initializes logtab and sqrttab: FastSqrt void inittabs() { sqrtab = new int[1<<(r/2)]; logtab = new int[1<<(r/2)]; for (int d = 0; d < r/2; d++) for (int k = 0; k < 1<<d; k++) logtab[1<<d+k] = d; int s = 1<<(r/4); // sqrt(2^(r/2)) for (int i = 0; i < 1<<(r/2); i++) { if ((s+1)*(s+1) <= i << (r/2)) s++; // sqrt increases sqrtab[i] = s; } } To summarize, the computations done by the i2b(i) method can be implemented √ in constant time on the word-RAM using O( n) extra memory to store the sqrttab and logtab arrays. These arrays can be rebuilt when n increases or decreases by a factor of 2, and the cost of this rebuilding can be amortized over the number of add(i, x) and remove(i) operations that caused the change in n in the same way that the cost of resize() is analyzed in the ArrayStack implementation. 2.7 Discussion and Exercises Most of the data structures described in this chapter are folklore. They can be found in implementations dating back over 30 years. For example, implementations of stacks, queues, and deques which generalize easily to the ArrayStack, ArrayQueue and ArrayDeque structures described here are discussed by Knuth [39, Section 2.2.2]. 41 2. Array-Based Lists 2.7. Discussion and Exercises Brodnik et al. [10] seem to have been the first to describe the RootishArrayStack √ and prove a n lower-bound like that in Section 2.6.2. They also present a different structure that uses a more sophisticated choice of block sizes in order to avoid computing square roots in the i2b(i) method. With their scheme, the block containing i is block blog(i+1)c, which is just the index of the leading 1 bit in the binary representation of i + 1. Some computer architectures provide an instruction for computing the index of the leading 1-bit in an integer. A structure related to the RootishArrayStack is the 2-level tiered-vector of Goodrich and Kloss [30]. This structure supports get(i, x) and set(i, x) in constant time and √ add(i, x) and remove(i) in O( n) time. These running times are similar to what can be achieved with the more careful implementation of a RootishArrayStack discussed in Exercise 2.9. Exercise 2.1. The List method addAll(i, c) inserts all elements of the Collection c into the list at position i. (The add(i, x) method is a special case where c = {x}.) Explain why, for the data structures in this chapter, it is not efficient to implement addAll(i, c) by repeated calls to add(i, x). Design and implement a more efficient implementation. Exercise 2.2. Design and implement a RandomQueue. This is an implementation of the Queue interface in which the remove() operation removes an element that is chosen uniformly at random among all the elements in the queue. The add(x) and remove() operations in a RandomQueue should take constant time. Exercise 2.3. Design and implement a Treque (triple-ended queue). This is a List imple- mentation in which get(i) and set(i, x) run in constant time and add(i, x) and remove(i) run in time O(1 + min{i, n − i, |n/2 − i|}) . With this running time, modifications are fast if they are near either end or near the middle of the list. Exercise 2.4. Implement a method rotate(r) that “rotates” a List so that list item i becomes list item (i + r) mod n. When run on an ArrayDeque, or a DualArrayDeque, rotate(r) should run in O(1 + min{r, n − r}). Exercise 2.5. Modify the ArrayDeque implementation so that the shifting done by add(i, x), remove(i), and resize() is done using System.arraycopy(s, i, d, j, n). Exercise 2.6. Modify the ArrayDeque implementation so that it does not use the % operator (which is expensive on some systems). Instead, it should make use of the fact that, if 42 2. Array-Based Lists 2.7. Discussion and Exercises a.length is a power of 2, then k%a.length=k&(a.length − 1). (Here, & is the bitwise-and operator.) Exercise 2.7. Design and implement a variant of ArrayDeque that does not do any modular arithmetic at all. Instead, all the data sits in a consecutive block, in order, inside an array. When the data overruns the beginning or the end of this array, a modified rebuild() operation is performed. The amortized cost of all operations should be the same as in an ArrayDeque. Hint: Making this work is really all about how a rebuild() operation is performed. You would like rebuild() to put the data structure into a state where the data cannot run off either end until at least n/2 operations have been performed. Test the performance of your implementation against the ArrayDeque. Optimize your implementation (by using System.arraycopy(a, i, b, i, n)) and see if you can get it to outperform the ArrayDeque implementation. √ Exercise 2.8. Design and implement a version of a RootishArrayStack that has only O( n) wasted space, but that can perform add(i, x) and remove(i, x) operations in O(1+min{i, n− i}) time. Exercise 2.9. Design and implement a version of a RootishArrayStack that has only √ O( n) wasted space, but that can perform add(i, x) and remove(i, x) operations in O(1 + √ min{ n, n − i}) time. (For an idea on how to do this, see Section 3.3.) Exercise 2.10. Design and implement a version of a RootishArrayStack that has only √ O( n) wasted space, but that can perform add(i, x) and remove(i, x) operations in O(1 + √ min{i, n, n − i}) time. (See Section 3.3 for ideas on how to achieve this.) 43 2. Array-Based Lists 2.7. Discussion and Exercises 44 Chapter 3 Linked Lists In this chapter, we continue to study implementations of the List interface, this time using pointer-based data structures rather than arrays. The structures in this chapter are made up of nodes that contain the list items. The nodes are linked together into a sequence using references (pointers). We first study singly-linked lists, which can implement Stack and (FIFO) Queue operations in constant time per operation. Linked lists have advantages and disadvantages relative to array-based implementa- tions of the List interface. The primary disadvantage is that we lose the ability to access any element using get(i) or set(i, x) in constant time. Instead, we have to walk through the list, one element at a time, until we reach the ith element. The primary advantage is that they are more dynamic: Given a reference to any list node u, we can delete u or insert a node adjacent to u in constant time. This is true no matter where u is in the list. 3.1 SLList: A Singly-Linked List An SLList (singly-linked list) is a sequence of Nodes. Each node u stores a data value u.x and a reference u.next to the next node in the sequence. For the last node w in the sequence, w.next = null SLList class Node { public: T x; Node *next; Node(T x0) { x = 0; next = NULL; } }; 45 3. Linked Lists 3.1. SLList: A Singly-Linked List head tail a b c d e head tail add(x) a b c d e x head tail remove() b c d e x head tail pop() c d e x head tail push(y) y c d e x Figure 3.1: A sequence of Queue (add(x) and remove()) and Stack (push(x) and pop()) operations on an SLList. For efficiency, an SLList uses variables head and tail to keep track of the first and last node in the sequence, as well as an integer n to keep track of the length of the sequence: SLList Node *head; Node *tail; int n; A sequence of Stack and Queue operations on an SLList is illustrated in Figure 3.1. An SLList can efficiently implement the Stack operations push() and pop() by adding and removing elements at the head of the sequence. The push() operation simply creates a new node u with data value x, sets u.next to the old head of the list and makes u the new head of the list. Finally, it increments n since the size of the SLList has increased by one: SLList T push(T x) { Node *u = new Node(x); u->next = head; head = u; if (n == 0) tail = u; n++; return x; } 46 3. Linked Lists 3.1. SLList: A Singly-Linked List The pop() operation, after checking that the SLList is not empty, removes the head by setting head = head.next and decrementing n. A special case occurs when the last element is being removed, in which case tail is set to null: SLList T pop() { if (n == 0) return NULL; T x = head->x; Node *u = head; head = head->next; delete u; if (--n == 0) tail = NULL; return x; } Clearly, both the push(x) and pop() operations run in O(1) time. 3.1.1 Queue Operations An SLList can also efficiently implement the FIFO queue operations add(x) and remove(). Removals are done from the head of the list, and are identical to the pop() operation: SLList T remove() { if (n == 0) return NULL; T x = head->x; Node *u = head; head = head->next; delete u; if (--n == 0) tail = NULL; return x; } Additions, on the other hand, are done at the tail of the list. In most cases, this is done by setting tail.next = u, where u is the newly created node that contains x. However, a special case occurs when n = 0, in which case tail = head = null. In this case, both tail and head are set to u. SLList bool add(T x) { Node *u = new Node(x); if (n == 0) { head = u; } else { tail->next = u; 47 3. Linked Lists 3.2. DLList: A Doubly-Linked List } tail = u; n++; return true; } Clearly, both add(x) and remove() take constant time. 3.1.2 Summary The following theorem summarizes the performance of an SLList: Theorem 3.1. An SLList implements the Stack and (FIFO) Queue interfaces. The push(x), pop(), add(x) and remove() operations run in O(1) time per operation. An SLList comes very close to implementing the full set of Deque operations. The only missing operation is removal from the tail of an SLList. Removing from the tail of an SLList is difficult because it requires updating the value of tail so that it points to the node w that precedes tail in the SLList; this is the node w such that w.next = tail. Unfortunately, the only way to get to w is by traversing the SLList starting at head and taking n − 2 steps. 3.2 DLList: A Doubly-Linked List A DLList (doubly-linked list) is very similar to an SLList except that each node u in a DLList has references to both the node u.next that follows it and the node u.prev that precedes it. DLList struct Node { T x; Node *prev, *next; }; When implementing an SLList, we saw that there were always some special cases to worry about. For example, removing the last element from an SLList or adding an element to an empty SLList requires special care so that head and tail are correctly updated. In a DLList, the number of these special cases increases considerably. Perhaps the cleanest way to take care of all these special cases in a DLList is to introduce a dummy node. This is a node that does not contain any data, but acts as a placeholder so that there are no special nodes; every node has both a next and a prev, with dummy acting as the node that follows 48 3. Linked Lists 3.2. DLList: A Doubly-Linked List dummy a b c d e Figure 3.2: A DLList containing a,b,c,d,e. the last node in the list and that precedes the first node in the list. In this way, the nodes of the list are (doubly-)linked into a cycle, as illustrated in Figure 3.2. DLList Node dummy; int n; DLList() { dummy.next = &dummy; dummy.prev = &dummy; n = 0; } Finding the node with a particular index in a DLList is easy; we can either start at the head of the list (dummy.next) and work forward, or start at the tail of the list (dummy.prev) and work backward. This allows us to reach the ith node in O(1 + min{i, n − i}) time: DLList Node* getNode(int i) { Node* p; if (i < n / 2) { p = dummy.next; for (int j = 0; j < i; j++) p = p->next; } else { p = &dummy; for (int j = n; j > i; j--) p = p->prev; } return (p); } The get(i) and set(i, x) operations are now also easy. We first find the ith node and then get or set its x value: 49 3. Linked Lists 3.2. DLList: A Doubly-Linked List DLList T get(int i) { return getNode(i)->x; } T set(int i, T x) { Node* u = getNode(i); T y = u->x; u->x = x; return y; } The running time of these operations is dominated by the time it takes to find the ith node, and is therefore O(1 + min{i, n − i}). 3.2.1 Adding and Removing If we have a reference to a node w in a DLList and we want to insert a node u before w, then this is just a matter of setting u.next = w, u.prev = w.prev, and then adjusting u.prev.next and u.next.prev. Thanks to the dummy node, there is no need to worry about w.prev or w.next not existing. DLList Node* addBefore(Node *w, T x) { Node *u = new Node; u->x = x; u->prev = w->prev; u->next = w; u->next->prev = u; u->prev->next = u; n++; return u; } Now, the list operation add(i, x) is trivial to implement. We find the ith node in the DLList and insert a new node u that contains x just before it. DLList void add(int i, T x) { addBefore(getNode(i), x); } The only non-constant part of the running time of add(i, x) is the time it takes to find the ith node (using getNode(i)). Thus, add(i, x) runs in O(1 + min{i, n − i}) time. 50 3. Linked Lists 3.2. DLList: A Doubly-Linked List Removing a node w from a DLList is easy. We need only adjust pointers at w.next and w.prev so that they skip over w. Again, the use of the dummy node eliminates the need to consider any special cases: DLList void remove(Node *w) { w->prev->next = w->next; w->next->prev = w->prev; delete w; n--; } Now the remove(i) operation is trivial. We find the node with index i and remove it: DLList T remove(int i) { Node *w = getNode(i); T x = w->x; remove(w); return x; } Again, the only expensive part of this operation is finding the ith node using getNode(i), so remove(i) runs in O(1 + min{i, n − i}) time. 3.2.2 Summary The following theorem summarizes the performance of a DLList: Theorem 3.2. A DLList implements the List interface. The get(i), set(i, x), add(i, x) and remove(i) operations run in O(1 + min{i, n − i}) time per operation. It is worth noting that, if we ignore the cost of the getNode(i) operation, then all operations on a DLList take constant time. Thus, the only expensive part of operations on a DLList is finding the relevant node. Once we have the relevant node, adding, removing, or accessing the data at that node takes only constant time. This is in sharp contrast to the array-based List implementations of Chapter 2; in those implementations, the relevant array item can be found in constant time. However, addition or removal requires shifting elements in the array and, in general, takes non- constant time. 51 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List For this reason, linked list structures are well-suited to applications where references to list nodes can be obtained through external means. For example, pointers to the nodes of a linked list could be stored in a USet. Then, to remove an item x from the linked list, the node that contains x can be found quickly using the Uset and the node can be removed from the list in constant time. 3.3 SEList: A Space-Efficient Linked List One of the drawbacks of linked lists (besides the time it takes to access elements that are deep within the list) is their space usage. Each node in a DLList requires an additional two references to the next and previous nodes in the list. Two thirds of the fields in a Node are dedicated to maintaining the list and only one third of the fields are for storing data! An SEList (space-efficient list) reduces this wasted space using a simple idea: Rather than store individual elements in a DLList, we store a block (array) containing several items. More precisely, an SEList is parameterized by a block size b. Each individual node in an SEList stores a block that can hold up to b + 1 elements. It will turn out, for reasons that become clear later, that it will be helpful if we can do Deque operations on each block. The data structure we choose for this is a BDeque (bounded deque), derived from the ArrayDeque structure described in Section 2.4. The BDeque differs from the ArrayDeque in one small way: When a new BDeque is created, the size of the backing array a is fixed at b + 1 and it never grows or shrinks. The important property of a BDeque is that it allows for the addition or removal of elements at either the front or back in constant time. This will be useful as elements are shifted from one block to another. SEList class BDeque : public ArrayDeque<T> { public: BDeque(int b) { n = 0; j = 0; array<int> z(b+1); a = z; } ~BDeque() { } // C++ Question: Why is this necessary? void add(int i, T x) { ArrayDeque<T>::add(i, x); } bool add(T x) { ArrayDeque<T>::add(size(), x); 52 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List return true; } void resize() {} }; An SEList is then a doubly-linked list of blocks: SEList class Node { public: BDeque d; Node *prev, *next; Node(int b) : d(b) { } }; SEList int n; Node dummy; 3.3.1 Space Requirements An SEList places very tight restrictions on the number of elements in a block: Unless a block is the last block, then that block contains at least b − 1 and at most b + 1 elements. This means that, if an SEList contains n elements, then it has at most n/(b − 1) + 1 = O(n/b) blocks. The BDeque for each block contains an array of length b + 1 but, for all blocks except the last, at most a constant amount of space is wasted in this array. The remaining memory used by a block is also constant. This means that the wasted space in an SEList √ is only O(b + n/b). By choosing an appropriate value of b (ideally in Θ( n)) we can make √ the space-overhead of an SEList approach the Ω( n) lower bound.1 3.3.2 Finding Elements The first challenge we face with an SEList is finding the list item with a given index i. Note that the location of an element consists of two parts: The node u that contains the block that contains the element as well as the index j of the element within its block. SEList class Location { public: 1 See Section 2.6.2 for a proof of this lower bound. 53 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List Node *u; int j; Location() { } Location(Node *u, int j) { this->u = u; this->j = j; } }; To find the block that contains a particular element, we proceed in the same way as in a DLList. We either start at the front of the list and traverse in the forward direction or at the back of the list and traverse backwards until we reach the node we want. The only difference is that, each time we move from one node to the next, we skip over a whole block of elements. SEList void getLocation(int i, Location &ell) { if (i < n / 2) { Node *u = dummy.next; while (i >= u->d.size()) { i -= u->d.size(); u = u->next; } ell.u = u; ell.j = i; } else { Node *u = &dummy; int idx = n; while (i < idx) { u = u->prev; idx -= u->d.size(); } ell.u = u; ell.j = i - idx; } } Remember that, with the exception of at most one block, each block contains at least b − 1 elements, so each step in our search gets us b − 1 elements closer to the element we are looking for. If we are searching forward, this means we reach the node we want after O(1 + i/b) steps. If we search backwards, we reach the node we want after O(1 + (n − i)/b) steps. The algorithm takes the smaller of these two quantities depending on the value of i, so the time to locate the item with index i is O(1 + min{i, n − i}/b). 54 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List Once we know how to locate the item with index i, the get(i) and set(i, x) oper- ations translate into getting or setting a particular index in the correct block: SEList T get(int i) { Location l; getLocation(i, l); return l.u->d.get(l.j); } T set(int i, T x) { Location l; getLocation(i, l); T y = l.u->d.get(l.j); l.u->d.set(l.j, x); return y; } These operations are dominated by the time it takes to locate the item, so they also run in O(1 + min{i, n − i}/b) time. 3.3.3 Adding an Element Things start to get complicated when adding elements to an SEList. Before considering the general case, we consider the easier operation, add(x), in which x is added to the end of the list. If the last block is full (or does not exist because there are no blocks yet), then we first allocate a new block and append it to the list of blocks. Now that we are sure that the last block exists and is not full, we append x to the last block. SEList void add(T x) { Node *last = dummy.prev; if (last == &dummy || last->d.size() == b+1) { last = addBefore(&dummy); } last->d.add(x); n++; } Things get more complicated when we add to the interior of the list using add(i, x). We first locate i to get the node u whose block contains the ith list item. The problem is that we want to insert x into u’s block, but we have to be prepared for the case where u’s block already contains b + 1 elements, so that it is full and there is no room for x. 55 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List ··· a b c d e f g h i j ··· ··· a x b c d e f g h i j ··· ··· a b c d e f g h ··· a x b c d e f g h ··· a b c d e f g h i j k l ··· ··· a x b c d e f g h i j k l ··· Figure 3.3: The three cases that occur during the addition of an item x in the interior of an SEList. (This SEList has block size b = 3.) Let u0 , u1 , u2 , . . . denote u, u.next, u.next.next, and so on. We explore u0 , u1 , u2 , . . . looking for a node that can provide space for x. Three cases can occur during our space exploration (see Figure 3.3): 1. We quickly (in r + 1 ≤ b steps) find a node ur whose block is not full. In this case, we perform r shifts of an element from one block into the next, so that the free space in ur becomes a free space in u0 . We can then insert x into u0 ’s block. 2. We quickly (in r + 1 ≤ b steps) run off the end of the list of blocks. In this case, we add a new empty block to the end of the list of blocks and proceed as in the first case. 3. After b steps we do not find any block that is not full. In this case, u0 , . . . , ub−1 is a sequence of b blocks that each contain b + 1 elements. We insert a new block ub at the end of this sequence and spread the original b(b + 1) elements so that each block of u0 , . . . , ub contains exactly b elements. Now u0 ’s block contains only b elements so it has room for us to insert x. SEList void add(int i, T x) { if (i == n) { add(x); return; 56 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List } Location l; getLocation(i, l); Node *u = l.u; int r = 0; while (r < b && u != &dummy && u->d.size() == b+1) { u = u->next; r++; } if (r == b) { // found b blocks each with b+1 elements spread(l.u); u = l.u; } if (u == &dummy) { // ran off the end of the list - add new node u = addBefore(u); } while (u != l.u) { // work backwards, shifting an element at each step u->d.add(0, u->prev->d.remove(u->prev->d.size()-1)); u = u->prev; } u->d.add(l.j, x); n++; } The running time of the add(i, x) operation depends on which of the three cases above occurs. Cases 1 and 2 involve examining and shifting elements through at most b blocks and take O(b) time. Case 3 involves calling the spread(u) method, which moves b(b + 1) elements and takes O(b2 ) time. If we ignore the cost of Case 3 (which we will account for later with amortization) this means that the total running time to locate i and perform the insertion of x is O(b + min{i, n − i}/b). 3.3.4 Removing an Element Removing an element, using the remove(i) method from an SEList is similar to adding an element. We first locate the node u that contains the element with index i. Now, we have to be prepared for the case where we cannot remove an element from u without causing u’s block to have size less than b − 1, which is not allowed. Again, let u0 , u1 , u2 , . . . denote u, u.next, u.next.next, We examine u0 , u1 , u2 , . . . in order looking for a node from which we can borrow an element to make the size of u0 ’s block larger than b − 1. There are three cases to consider (see Figure 3.4): 1. We quickly (in r + 1 ≤ b steps) find a node whose block contains more than b − 1 57 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List ··· a b c d e f g ··· ··· a c d e f g ··· ··· a b c d e f ··· a c d e f ··· a b c d e f ··· ··· a c d e f ··· Figure 3.4: The three cases that occur during the removal of an item x in the interior of an SEList. (This SEList has block size b = 3.) elements. In this case, we perform r shifts of an element from one block into the previous, so that the extra element in ur becomes an extra element in u0 . We can then remove the appropriate element from u0 ’s block. 2. We quickly (in r + 1 ≤ b steps) run off the end of the list of blocks. In this case, ur is the last block, and there is no requirement that ur ’s block contain at least b − 1 elements. Therefore, we proceed as above, borrowing an element from ur to make an extra element in u0 . If this causes ur ’s block to become empty, then we remove it. 3. After b steps we do not find any block containing more than b − 1 elements. In this case, u0 , . . . , ub−1 is a sequence of b blocks that each contain b−1 elements. We gather these b(b − 1) elements into u0 , . . . , ub−2 so that each of these b − 1 blocks contains exactly b elements and we remove ub−1 , which is now empty. Now u0 ’s block contains b elements so we can remove the appropriate element from it. SEList T remove(int i) { Location l; getLocation(i, l); T y = l.u->d.get(l.j); Node *u = l.u; int r = 0; while (r < b && u != &dummy && u->d.size() == b - 1) { 58 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List u = u->next; r++; } if (r == b) { // found b blocks each with b-1 elements gather(l.u); } u = l.u; u->d.remove(l.j); while (u->d.size() < b - 1 && u->next != &dummy) { u->d.add(u->next->d.remove(0)); u = u->next; } if (u->d.size() == 0) remove(u); n--; return y; } Like the add(i, x) operation, the running time of the remove(i) operation is O(b + min{i, n − i}/b) if we ignore the cost of the gather(u) method that occurs in Case 3. 3.3.5 Amortized Analysis of Spreading and Gathering Next, we consider the cost of the gather(u) and spread(u) methods that may be executed by the add(i, x) and remove(i) methods. For completeness, here they are: SEList void spread(Node *u) { Node *w = u; for (int j = 0; j < b; j++) { w = w->next; } w = addBefore(w); while (w != u) { while (w->d.size() < b) w->d.add(0, w->prev->d.remove(w->prev->d.size()-1)); w = w->prev; } } SEList void gather(Node *u) { Node *w = u; for (int j = 0; j < b-1; j++) { while (w->d.size() < b) w->d.add(w->next->d.remove(0)); 59 3. Linked Lists 3.3. SEList: A Space-Efficient Linked List w = w->next; } remove(w); } The running time of each of these methods is dominated by the two nested loops. Both the inner loop and outer loop execute at most b + 1 times, so the total running time of each of these methods is O((b + 1)2 ) = O(b2 ). However, the following lemma shows that these methods execute on at most one out of every b calls to add(i, x) or remove(i). Lemma 3.1. If an empty SEList is created and any sequence of m ≥ 1 calls to add(i, x) and remove(i) are performed, then the total time spent during all calls to spread() and gather() is O(bm). Proof. We will use the potential method of amortized analysis. We say that a node u is fragile if u’s block does not contain b elements (so that u is either the last node, or contains b − 1 or b + 1 elements). Any node whose block contains b elements is rugged. Define the potential of an SEList as the number of fragile nodes it contains. We will consider only the add(i, x) operation and its relation to the number of calls to spread(u). The analysis of remove(i) and gather(u) is identical. Notice that, if Case 1 occurs during the add(i, x) method, then only one node, ur has the size of its block changed. Therefore, at most one node, namely ur , goes from being rugged to being fragile. If Case 2 occurs, then a new node is created, and this node is fragile, but no other node changes sizes, so the number of fragile nodes increases by one. Thus, in either Case 1 or Case 2 the potential of the SEList increases by at most 1. Finally, if Case 3 occurs, it is because u0 , . . . , ub−1 are all fragile nodes. Then spread(u0 ) is called and these b fragile nodes are replaced with b + 1 rugged nodes. Finally, x is added to u0 ’s block, making u0 fragile. In total the potential decreases by b − 1. In summary, the potential starts at 0 (there are no nodes in the list). Each time Case 1 or Case 2 occurs, the potential increases by at most 1. Each time Case 3 occurs, the potential decreases by b − 1. The potential (which counts the number of fragile nodes) is never less than 0. We conclude that, for every occurrence of Case 3, there are at least b − 1 occurrences of Case 1 or Case 2. Thus, for every call to spread(u) there are at least b calls to add(i, x). This completes the proof. 60 3. Linked Lists 3.4. Discussion and Exercises 3.3.6 Summary The following theorem summarizes the performance of the SEList data structure: Theorem 3.3. An SEList implements the List interface. Ignoring the cost of calls to spread(u) and gather(u), an SEList with block size b supports the operations • get(i) and set(i, x) in O(1 + min{i, n − i}/b) time per operation; and • add(i, x) and remove(i) in O(b + min{i, n − i}/b) time per operation. Furthermore, beginning with an empty SEList, any sequence of m add(i, x) and remove(i) operations results in a total of O(bm) time spent during all calls to spread(u) and gather(u). The space (measured in words)2 used by an SEList that stores n elements is n + O(b + n/b). The SEList is a tradeoff between an ArrayList and a DLList where the relative mix of these two structures depends on the block size b. At the extreme b = 2, each SEList node stores at most 3 values, which is really not much different than a DLList. At the other extreme, b > n, all the elements are stored in a single array, just like in an ArrayList. In between these two extremes lies a tradeoff between the time it takes to add or remove a list item and the time it takes to locate a particular list item. 3.4 Discussion and Exercises Both singly-linked and doubly-linked lists are folklore, having been used in programs for over 40 years. They are discussed, for example, by Knuth [39, Sections 2.2.3–2.2.5]. Even the SEList data structure seems to be a well-known data structures exercise. Exercise 3.1. Why is it not possible, in an SLList to use a dummy node to avoid all the special cases that occur in the operations push(x), pop(), add(x), and remove()? Exercise 3.2. Describe and implement the List operations get(i), set(i, x), add(i, x) and remove(i) on an SLList. Each of these operations should run in O(1 + i) time. 2 Recall Section 1.3 for a discussion of how memory is measured. 61 3. Linked Lists 3.4. Discussion and Exercises 62 Chapter 4 Skiplists In this chapter, we discuss a beautiful data structure: the skiplist, which has a variety of applications. Using a skiplist we can implement a List that is fast for all the operations get(i), set(i, x), add(i, x), and remove(i). We can also implement an SSet in which all operations run in O(log n) expected time. Skiplists rely on randomization for their efficiency. In particular, a skiplist uses random coin tosses when an element is inserted to determine the height of that element. The performance of skiplists is expressed in terms of expected running times and lengths of paths. This expectation is taken over the random coin tosses used by the skiplist. In the implementation, the random coin tosses used by a skiplist are simulated using a pseudo- random number (or bit) generator. 4.1 The Basic Structure Conceptually, a skiplist is a sequence of singly-linked lists L0 , . . . , Lh , where each Lr contains a subset of the items in Lr−1 . We start with the input list L0 that contains n items and construct L1 from L0 , L2 from L1 , and so on. The items in Lr are obtained by tossing a coin for each element, x, in Lr−1 and including x in Lr if the coin comes up heads. This process ends when we create a list Lr that is empty. An example of a skiplist is shown in Figure 4.1. For an element, x, in a skiplist, we call the height of x the largest value r such that x appears in Lr . Thus, for example, elements that only appear in L0 have height 0. Notice that the height of x corresponds to the following experiment: Toss a coin repeatedly until the first time it comes up tails. How many times did it come up heads? The answer, not surprisingly, is that the expected height of a node is 1. (We expect to toss the coin twice before getting tails, but we don’t count the last toss.) The height of a skiplist is the height of its tallest node. 63 4. Skiplists 4.1. The Basic Structure L5 L4 L3 L2 L1 L0 0 1 2 3 4 5 6 sentinel Figure 4.1: A skiplist containing seven elements. L5 L4 L3 L2 L1 L0 0 1 2 3 4 5 6 sentinel Figure 4.2: The search path for the node containing 4 in a skiplist. At the head of every list is a special node, called the sentinel, that acts as a dummy node for the list. The key property of skiplists is that there is a short path, called the search path, from the sentinel in Lh to every node in L0 . Remembering how to construct a search path for a node, u, is easy (see Figure 4.2) : Start at the top left corner of your skiplist (the sentinel in Lh ) and always go right unless that would overshoot u, in which case you should take a step down into the list below. More precisely, to construct the search path for the node u in L0 we start at the sentinel, w, in Lh . Next, we examine w.next. If w.next contains an item that appears before u in L0 , then we set w = w.next. Otherwise, we move down and continue the search at the occurrence of w in the list Lh−1 . We continue this way until we reach the predecessor of u in L0 . The following result, which we will prove in Section 4.4, shows that the search path is quite short: Lemma 4.1. The expected length of the search path for any node, u, in L0 is at most 2 log n + O(1) = O(log n). 64 4. Skiplists 4.2. SkiplistSSet: An Efficient SSet Implementation A space-efficient way to implement a Skiplist is to define a Node, u, as consisting of a data value, x, and an array, next, of pointers, where u.next[i] points to u’s successor in the list Li . In this way, the data, x, in a node is stored only once, even though x may appear in several lists. SkiplistSSet struct Node { T x; int height; // length of next Node *next[]; }; The next two sections of this chapter discuss two different applications of skiplists. In each of these applications, L0 stores the main structure (a list of elements or a sorted set of elements). The primary difference between these structures is in how a search path is navigated; in particular, they differ in how they decide if a search path should go down into Lr−1 or go right within Lr . 4.2 SkiplistSSet: An Efficient SSet Implementation A SkiplistSSet uses a skiplist structure to implement the SSet interface. When used this way, the list L0 stores the elements of the SSet in sorted order. The find(x) method works by following the search path for the smallest value y such that y ≥ x: SkiplistSSet Node* findPredNode(T x) { Node *u = sentinel; int r = h; while (r >= 0) { while ( u->next[r] != NULL && compare(u->next[r]->x, x) < 0) u = u->next[r]; // go right in list r r--; // go down into list r-1 } return u; } T find(T x) { Node *u = findPredNode(x); return u->next[0] == NULL ? NULL : u->next[0]->x; } Following the search path for y is easy: when situated at some node, u, in Lr , we look right to u.next[r].x. If x > u.next[r].x, then we take a step to the right in Lr , otherwise 65 4. Skiplists 4.2. SkiplistSSet: An Efficient SSet Implementation we move down into Lr−1 . Each step (right or down) in this search takes only constant time so, by Lemma 4.1, the expected running time of find(x) is O(log n). Before we can add an element to a SkipListSSet, we need a method to simulate tossing coins to determine the height, k, of a new node. We do this by picking a random integer, z, and counting the number of trailing 1s in the binary representation of z:1 SkiplistSSet int pickHeight() { int z = rand(); int k = 0; int m = 1; while ((z & m) != 0) { k++; m <<= 1; } return k; } To implement the add(x) method in a SkiplistSSet we search for x and then splice x into a few lists L0 ,. . . ,Lk , where k is selected using the pickHeight() method. The easiest way to do this is to use an array, stack, that keeps track of the nodes at which the search path goes down from some list Lr into Lr−1 . More precisely, stack[r] is the node in Lr where the search path proceeded down into Lr−1 . The nodes that we modify to insert x are precisely the nodes stack[0], . . . , stack[k]. The following code implements this algorithm for add(x): SkiplistSSet bool add(T x) { Node *u = sentinel; int r = h; int comp = 0; while (r >= 0) { while (u->next[r] != NULL && (comp = compare(u->next[r]->x, x)) < 0) u = u->next[r]; if (u->next[r] != NULL && comp == 0) return false; stack[r--] = u; // going down, store u } Node *w = newNode(x, pickHeight()); while (h < w->height) 1 This method does not exactly replicate the coin-tossing experiment since the value of k will always be less than the number of bits in an int. However, this will have negligible impact unless the number of elements in the structure is much greater than 232 = 4294967296. 66 4. Skiplists 4.2. SkiplistSSet: An Efficient SSet Implementation 0 1 2 3 3.5 4 5 6 sentinel add(3.5) Figure 4.3: Adding the node containing 3.5 to a skiplist. The nodes stored in stack are highlighted. stack[++h] = sentinel; // increasing height of skiplist for (int i = 0; i < w->height; i++) { w->next[i] = stack[i]->next[i]; stack[i]->next[i] = w; } n++; return true; } Removing an element, x, is done in a similar way, except that there is no need for stack to keep track of the search path. The removal can be done as we are following the search path. We search for x and each time the search moves downward from a node u, we check if u.next.x = x and if so, we splice u out of the list: SkiplistSSet bool remove(T x) { bool removed = false; Node *u = sentinel, *del; int r = h; int comp = 0; while (r >= 0) { while (u->next[r] != NULL && (comp = compare(u->next[r]->x, x)) < 0) { u = u->next[r]; } if (u->next[r] != NULL && comp == 0) { removed = true; del = u->next[r]; u->next[r] = u->next[r]->next[r]; if (u == sentinel && u->next[r] == NULL) h--; // skiplist height has gone down } 67 4. Skiplists 4.3. SkiplistList: An Efficient Random-Access List Implementation 0 1 2 3 4 5 6 sentinel remove(3) Figure 4.4: Removing the node containing 3 from a skiplist. r--; } if (removed) { delete del; n--; } return removed; } 4.2.1 Summary The following theorem summarizes the performance of skiplists when used to implement sorted sets: Theorem 4.1. A SkiplistSSet implements the SSet interface. A SkiplistSSet supports the operations add(x), remove(x), and find(x) in O(log n) expected time per operation. 4.3 SkiplistList: An Efficient Random-Access List Implementation A SkiplistList implements the List interface on top of a skiplist structure. In a SkiplistList, L0 contains the elements of the list in the order they appear in the list. Just like with a SkiplistSSet, elements can be added, removed, and accessed in O(log n) time. For this to be possible, we need a way to follow the search path for the ith element in L0 . The easiest way to do this is to define the notion of the length of an edge in some list, Lr . We define the length of every edge in L0 as 1. The length of an edge, e, in Lr , r > 0, is defined as the sum of the lengths of the edges below e in Lr−1 . Equivalently, the length of e is the number of edges in L0 below e. See Figure 4.5 for an example of a skiplist with the lengths of its edges shown. Since the edges of skiplists are stored in arrays, the lengths can be stored the same way: 68 4. Skiplists 4.3. SkiplistList: An Efficient Random-Access List Implementation 5 L5 5 L4 3 2 L3 3 1 1 L2 3 1 1 1 1 L1 1 1 1 1 1 1 1 L0 0 1 2 3 4 5 6 sentinel Figure 4.5: The lengths of the edges in a skiplist. SkiplistList struct Node { T x; int height; // length of next int *length; Node **next; }; The useful property of this definition of length is that, if we are currently at a node that is at position j in L0 and we follow an edge of length `, then we move to a node whose position, in L0 , is j + `. In this way, while following a search path, we can keep track of the position, j, of the current node in L0 . When at a node, u, in Lr , we go right if j plus the length of the edge u.next[r] is less than i, otherwise we go down into Lr−1 . SkiplistList Node* findPred(int i) { Node *u = sentinel; int r = h; int j = -1; // the index of the current node in list 0 while (r >= 0) { while (u->next[r] != NULL && j + u->length[r] < i) { j += u->length[r]; u = u->next[r]; } r--; } return u; } SkiplistList T get(int i) { return findPred(i)->next[0]->x; 69 4. Skiplists 4.3. SkiplistList: An Efficient Random-Access List Implementation 56 56 3 232 1 3 1 121 1 3 1 121 1 1 1 1 1 1 1 121 1 1 1 0 1 2 3 x 4 5 6 sentinel add(4,x) Figure 4.6: Adding an element to a SkiplistList. } T set(int i, T x) { Node *u = findPred(i)->next[0]; T y = u->x; u->x = x; return y; } Since the hardest part of the operations get(i) and set(i, x) is finding the ith node in L0 , these operations run in O(log n) time. Adding an element to a SkiplistList at a position, i, is fairly straightforward. Unlike in a SkiplistSSet, we are sure that a new node will actually be added, so we can do the addition at the same time as we search for the new node’s location. We first pick the height, k, of the newly inserted node, w, and then follow the search path for i. Anytime the search path moves down from Lr with r ≤ k, we splice w into Lr . The only extra care needed is to ensure that the lengths of edges are updated properly. See Figure 4.6. Note that, each time the search path goes down at a node, u, in Lr , the length of the edge u.next[r] increases by one, since we are adding an element below that edge at position i. Splicing the node w between two nodes, u and z, works as shown in Figure 4.7. While following the search path we are already keeping track of the position, j, of u in L0 . Therefore, we know that the length of the edge from u to w is i − j. We can also deduce the length of the edge from w to z from the length, `, of the edge from u to z. Therefore, we can splice in w and update the lengths of the edges in constant time. This sounds more complicated than it actually is and the code is actually quite simple: 70 4. Skiplists 4.3. SkiplistList: An Efficient Random-Access List Implementation u z ` j `+1 u w z i−j ` + 1 − (i − j) j i Figure 4.7: Updating the lengths of edges while splicing a node w into a skiplist. SkiplistList void add(int i, T x) { Node *w = newNode(x, pickHeight()); if (w->height > h) h = w->height; add(i, w); } SkiplistList Node* add(int i, Node *w) { Node *u = sentinel; int k = w->height; int r = h; int j = -1; // index of u while (r >= 0) { while (u->next[r] != NULL && j + u->length[r] < i) { j += u->length[r]; u = u->next[r]; } u->length[r]++; // to account for new node in list 0 if (r <= k) { w->next[r] = u->next[r]; u->next[r] = w; w->length[r] = u->length[r] - (i - j); u->length[r] = i - j; } r--; } n++; return u; } By now, the implementation of the remove(i) operation in a SkiplistList should 71 4. Skiplists 4.3. SkiplistList: An Efficient Random-Access List Implementation 54 L5 54 L4 3 21 L3 1 3 1 1 L2 1 3 1 1 1 1 L1 1 1 1 1 1 1 1 1 L0 0 1 2 3 4 5 6 sentinel remove(3) Figure 4.8: Removing an element from a SkiplistList. be obvious. We follow the search path for the node at position i. Each time the search path takes a step down from a node, u, at level r we decrement the length of the edge leaving u at that level. We also check if u.next[r] is the element of rank i and, if so, splice it out of the list at that level. An example is shown in Figure 4.8. SkiplistList T remove(int i) { T x = NULL; Node *u = sentinel, *del; int r = h; int j = -1; // index of node u while (r >= 0) { while (u->next[r] != NULL && j + u->length[r] < i) { j += u->length[r]; u = u->next[r]; } u->length[r]--; // for the node we are removing if (j + u->length[r] + 1 == i && u->next[r] != NULL) { x = u->next[r]->x; u->length[r] += u->next[r]->length[r]; del = u->next[r]; u->next[r] = u->next[r]->next[r]; if (u == sentinel && u->next[r] == NULL) h--; } r--; } deleteNode(del); n--; return x; } 72 4. Skiplists 4.4. Analysis of Skiplists 4.3.1 Summary The following theorem summarizes the performance of the SkiplistList data structure: Theorem 4.2. A SkiplistList implements the List interface. A SkiplistList supports the operations get(i), set(i, x), add(i, x), and remove(i) in O(log n) expected time per operation. 4.4 Analysis of Skiplists In this section, we analyze the expected height, size, and length of the search path in a skiplist. This section requires a background in basic probability. Several proofs are based on the following basic observation about coin tosses. Lemma 4.2. Let T be the number of times a fair coin is tossed up to and including the first time the coin comes up heads. Then E[T ] = 2. Proof. Suppose we stop tossing the coin the first time it comes up heads. Define the indicator variable ( 0 if the coin is tossed less than i times Ii = 1 if the coin is tossed i or more times Note that Ii = 1 if and only if the first i − 1 coin tosses are tails, so E[Ii ] = Pr{Ii = 1} = P 1/2i−1 . Observe that T , the total number of coin tosses, can be written as T = ∞ i=1 Ii . Therefore, "∞ # X E[T ] = E Ii i=1 ∞ X = E [Ii ] i=1 X∞ = 1/2i−1 i=1 = 1 + 1/2 + 1/4 + 1/8 + · · · =2 . The next two lemmata tell us that skiplists have linear size: Lemma 4.3. The expected number of nodes in a skiplist containing n elements, not includ- ing occurrences of the sentinel, is 2n. 73 4. Skiplists 4.4. Analysis of Skiplists Proof. The probability that any particular element, x, is included in list Lr is 1/2r , so the expected number of nodes in Lr is n/2r . Therefore, the total number of nodes in all lists is ∞ X n/2r = n(1 + 1/2 + 1/4 + 1/8 + · · · ) = 2n . r=0 Lemma 4.4. The expected height of a skiplist containing n elements is at most log n + 2. Proof. For each r ∈ {1, 2, 3, . . . , ∞}, define the indicator random variable ( 0 if Lr is empty Ir = 1 if Lr is non-empty The height, h, of the skiplist is then given by ∞ X h= Ir . i=1 Note that Ir is never more than the length, |Lr |, of Lr , so E[Ir ] ≤ E[|Lr |] = n/2r . Therefore, we have " ∞ # X E[h] = E Ir r=1 ∞ X = E[Ir ] r=1 blog nc ∞ X X = E[Ir ] + E[Ir ] r=1 r=blog nc+1 blog nc ∞ X X ≤ 1+ n/2r r=1 r=blog nc+1 ∞ X r ≤ log n + 1/2 r=0 = log n + 2 . Lemma 4.5. The expected number of nodes in a skiplist containing n elements, including all occurrences of the sentinel, is 2n + O(log n). 74 4. Skiplists 4.4. Analysis of Skiplists Proof. By Lemma 4.3, the expected number of nodes, not including the sentinel, is 2n. The number of occurrences of the sentinel is equal to the height, h, of the skiplist so, by Lemma 4.4 the expected number of occurrences of the sentinel is at most log n + 2 = O(log n). Lemma 4.6. The expected length of a search path in a skiplist is at most 2 log n + O(1). Proof. The easiest way to see this is to consider the reverse search path for a node, x. This path starts at the predecessor of x in L0 . At any point in time, if the path can go up a level, then it does. If it cannot go up a level then it goes left. Observe that the reverse search path for x is identical to the search path for x, except that it is reversed. The number of nodes that the reverse search path visits at a particular level, r, is related to the following experiment: Toss a coin. If the coin comes up heads then go up and stop, otherwise go left and repeat the experiment. The number of coin tosses before the heads then represents the number of steps to the left that a reverse search path takes at a particular level.2 Lemma 4.2 tells us that the expected number of coin tosses before the first heads is 1. Let Sr denote the number of steps the forward search path takes at level r that go to the right. We have just argued that E[Sr ] ≤ 1. Furthermore, Sr ≤ |Lr |, since we can’t take more steps in Lr than the length of Lr , so E[Sr ] ≤ E[|Lr |] = n/2r . We can now finish as in the proof of Lemma 4.4. Let S be the length of the search path for some node, u, in a skiplist, and let h be the height of the skiplist. Then " ∞ # X E[S] = E h + Sr r=0 ∞ X = E[h] + E[Sr ] r=0 blog nc ∞ X X = E[h] + E[Sr ] + E[Sr ] r=0 r=blog nc+1 blog nc ∞ X X ≤ E[h] + 1+ n/2r r=0 r=blog nc+1 2 Note that this might overcount the number of steps to the left, since the experiment should end either at the first heads or when the search path reaches the sentinel, whichever comes first. This is not a problem since the lemma is only stating an upper bound. 75 4. Skiplists 4.5. Discussion and Exercises blog nc ∞ X X ≤ E[h] + 1+ 1/2r r=0 r=0 blog nc ∞ X X ≤ E[h] + 1+ 1/2r r=0 r=0 ≤ E[h] + log n + 3 ≤ 2 log n + 5 . The following theorem summarizes the results in this section: Theorem 4.3. A skiplist containing n elements has expected size O(n) and the expected length of the search path for any particular element is at most 2 log n + O(1). 4.5 Discussion and Exercises Skiplists were introduced by Pugh [50] who also presented a number of applications of skiplists [49]. Since then they have been studied extensively. Several researchers have done very precise analysis of the expected length and variance in length of the search path for the ith element in a skiplist [38, 37, 47]. Deterministic versions [44], biased versions [6, 21], and self-adjusting versions [9] of skiplists have all been developed. Skiplist implementations have been written for various languages and frameworks and have seen use in open-source database systems [58, 52]. A variant of skiplists is used in the HP-UX operating system kernel’s process management structures [36]. Exercise 4.1. Show that, during an add(x) or a remove(x) operation, the expected number of pointers in the structure that get changed is constant. Exercise 4.2. Suppose that, instead of promoting an element from Li−1 into Li based on a coin toss, we promote it with some probability p, 0 < p < 1. Show that the expected length of the search path in this case is at most (1/p) log1/p n + O(1). What is the value of p that minimizes this expression? What is the expected height of the skiplist? What is the expected number of nodes in the skiplist? Exercise 4.3. Design and implement a find(x) method for SkiplistSSet that avoids locally- redundant comparisons; these are comparisons that have already been done and occur be- cause u.next[r] = u.next[r − 1]. Analyze the expected number of comparisons done by your modified find(x) method. Exercise 4.4. Design and implement a version of a skiplist that implements the SSet inter- face, but also allows fast access to elements by rank. That is, it also supports the function 76 4. Skiplists 4.5. Discussion and Exercises get(i), which returns the element whose rank is i in O(log n) expected time. (The rank of an element x in an SSet is the number of elements in the SSet that are less than x.) Exercise 4.5. Using the ideas from the space-efficient linked-list, SEList, design and imple- ment a space-efficient SSet, SESSet. Do this by storing the data, in order, in an SEList and then storing the blocks of this SEList in an SSet. If the original SSet implementation uses O(n) space to store n elements, then the SESSet will use enough space for n elements plus O(n/b + b) wasted space. Exercise 4.6. Using an SSet as your underlying structure, design and implement an appli- cation that reads a (large) text file and allow you to search, interactively, for any substring contained in the text. Hint 1: Every substring is a prefix of some suffix, so it suffices to store all suffixes of the text file. Hint 2: Any suffix can be represented compactly as a single integer indicating where the suffix begins in the text. Test your application on some large texts like some of the books available at Project Guten- berg [1]. 77 4. Skiplists 4.5. Discussion and Exercises 78 Chapter 5 Hash Tables Hash tables are an efficient method of storing a small number, n, of integers from a large range U = {0, . . . , 2w − 1}. The term hash table includes a broad range of data structures. This chapter focuses on one of the most common implementations of hash tables, namely hashing with chaining. Very often hash tables store data that are not integers. In this case, an integer hash code is associated with each data item and this hash code is used in the hash table. The second part of this chapter discusses how such hash codes are generated. Some of the methods used in this chapter require random choices of integers in some specific range. In the code samples, some of these “random” integers are hard-coded constants. These constants were obtained using random bits generated from atmospheric noise. 5.1 ChainedHashTable: Hashing with Chaining A ChainedHashTable data structure uses hashing with chaining to store data as an array, t, of lists. An integer, n, keeps track of the total number of items in all lists: ChainedHashTable array<List> t; int n; The hash value of a data item x, denoted hash(x) is a value in the range {0, . . . , t.length − 1}. All items with hash value i are stored in the list at t[i]. To ensure that lists don’t get too long, we maintain the invariant n ≤ t.length so that the average number of elements stored in one of these lists is n/t.length ≤ 1. 79 5. Hash Tables 5.1. ChainedHashTable: Hashing with Chaining To add an element, x, to the hash table, we first check if the length of t needs to be increased and, if so, we grow t. With this out of the way we hash x to get an integer, i, in the range {0, . . . , t.length − 1} and we append x to the list t[i]: ChainedHashTable bool add(T x) { if (find(x) != null) return false; if (n+1 > t.length) resize(); t[hash(x)].add(x); n++; return true; } Growing the table, if necessary, involves doubling the length of t and reinserting all ele- ments into the new table. This is exactly the same strategy used in the implementation of ArrayStack and the same result applies: The cost of growing is only constant when amortized over a sequence of insertions (see Lemma 2.1 on page 20). Besides growing, the only other work done when adding x to a ChainedHashTable involves appending x to the list t[hash(x)]. For any of the list implementations described in Chapters 2 or 3, this takes only constant time. To remove an element x from the hash table we iterate over the list t[hash(x)] until we find x so that we can remove it: ChainedHashTable T remove(T x) { int j = hash(x); for (int i = 0; i < t[j].size(); i++) { T y = t[j].get(i); if (x == y) { t[j].remove(i); n--; return y; } } return null; } This takes O(nhash(x) )time, where ni denotes the length of the list stored at t[i]. Searching for the element x in a hash table is similar. We perform a linear search on the list t[hash(x)]: ChainedHashTable T find(T x) { int j = hash(x); 80 5. Hash Tables 5.1. ChainedHashTable: Hashing with Chaining for (int i = 0; i < t[j].size(); i++) if (x == t[j].get(i)) return t[j].get(i); return null; } Again, this takes time proportional to the length of the list t[hash(x)]. The performance of a hash table depends critically on the choice of the hash function. A good hash function will spread the elements evenly among the t.length lists, so that the expected size of the list t[hash(x)] is O(n/t.length) = O(1). On the other hand, a bad hash function will hash all values (including x) to the same table location, in which case the size of the list t[hash(x)] will be n. In the next section we describe a good hash function. 5.1.1 Multiplicative Hashing Multiplicative hashing is an efficient method of generating hash values based on modular arithmetic (discussed in Section 2.3) and integer division. It uses the div operator, which calculates the integral part of a quotient, while discarding the remainder. Formally, for any integers a ≥ 0 and b ≥ 1, a div b = ba/bc. In multiplicative hashing, we use a hash table of size 2d for some integer d (called the dimension). The formula for hashing an integer x ∈ {0, . . . , 2w − 1} is hash(x) = ((z · x) mod 2w ) div 2w−d . Here, z is a randomly chosen odd integer in {1, . . . , 2w − 1}. This hash function can be realized very efficiently by observing that, by default, operations on integers are already done modulo 2w where w is the number of bits in an integer. (See Figure 5.1.) Furthermore, integer division by 2w−d is equivalent to dropping the rightmost w − d bits in a binary representation (which is implemented by shifting the bits right by w − d). In this way, the code that implements the above formula is simpler than the formula itself: ChainedHashTable int hash(T x) { return ((unsigned)(z * hashCode(x))) >> (w-d); } The following lemma, whose proof is deferred until later in this section, shows that multiplicative hashing does a good job of avoiding collisions: Lemma 5.1. Let x and y be any two values in {0, . . . , 2w −1} with x 6= y. Then Pr{hash(x) = hash(y)} ≤ 2/2d . 81 5. Hash Tables 5.1. ChainedHashTable: Hashing with Chaining 2w (4294967296) 100000000000000000000000000000000 z (4102541685) 11110100100001111101000101110101 x (42) 00000000000000000000000000101010 z·x 10100000011110010010000101110100110010 (z · x) mod 2w 00011110010010000101110100110010 ((z · x) mod 2w ) div 2w−d 00011110 Figure 5.1: The operation of the multiplicative hash function with w = 32 and d = 8. With Lemma 5.1, the performance of remove(x), and find(x) are easy to analyze: Lemma 5.2. For any data value x, the expected length of the list t[hash(x)] is at most nx + 2, where nx is the number of occurrences of x in the hash table. Proof. Let S be the (multi-)set of elements stored in the hash table that are not equal to x. For an element y ∈ S, define the indicator variable ( 1 if hash(x) = hash(y) Iy = 0 otherwise and notice that, by Lemma 5.1, E[Iy ] ≤ 2/2d = 2/t.length. The expected length of the list t[hash(x)] is given by X E [t[hash(x)].size()] = E nx + Iy y∈S X = nx + E[Iy ] y∈S X ≤ nx + 2/t.length y∈S X ≤ nx + 2/n y∈S ≤ nx + (n − nx )2/n ≤ nx + 2 , as required. Now, we want to prove Lemma 5.1, but first we need a result from number theory. P In the following proof, we use the notation (br , . . . , b0 )2 to denote ri=0 bi 2i , where each bi is 82 5. Hash Tables 5.1. ChainedHashTable: Hashing with Chaining a bit, either 0 or 1. In other words, (br , . . . , b0 )2 is the integer whose binary representation is given by br , . . . , b0 . We use ? to denote a bit of unknown value. Lemma 5.3. Let S be the set of odd integers in {1, . . . , 2w − 1}, Let q and i be any two elements in S. Then there is exactly one value z ∈ S such that zq mod 2w = i. Proof. Since the number of choices for z and i is the same, it is sufficient to prove that there is at most one value z ∈ S that satisfies zq mod 2w = i. Suppose, for the sake of contradiction, that there are two such values z and z0 , with z > z0 . Then zq mod 2w = z0 q mod 2w = i So (z − z0 )q mod 2w = 0 But this means that (z − z0 )q = k2w (5.1) for some integer k. Thinking in terms of binary numbers, we have (z − z0 )q = k · (1, 0, . . . , 0)2 , | {z } w so that the w trailing bits in the binary representation of (z − z0 )q are all 0’s. Furthermore k 6= 0 since q 6= 0 and z − z0 6= 0. Since q is odd, it has no trailing 0’s in its binary representation: q = (?, . . . , ?, 1)2 . Since |z − z0 | < 2w , z − z0 has fewer than w trailing 0’s in its binary representation: z − z0 = (?, . . . , ?, 1, 0, . . . , 0)2 . | {z } <w Therefore, the product (z − z0 )q has fewer than w trailing 0’s in its binary representation: (z − z0 )q = (?, · · · , ?, 1, 0, . . . , 0)2 . | {z } <w Therefore (z − z0 )q cannot satisfy (5.1), yielding a contradiction and completing the proof. 83 5. Hash Tables 5.1. ChainedHashTable: Hashing with Chaining The utility of Lemma 5.3 comes from the following observation: If z is chosen uniformly at random from S, then zt is uniformly distributed over S. In the following proof, it helps to think of the binary representation of z, which consists of w − 1 random bits followed by a 1. Proof of Lemma 5.1. First we note that the condition hash(x) = hash(y) is equivalent to the statement “the highest-order d bits of zx mod 2w and the highest-order d bits of zy mod 2w are the same.” A necessary condition of that statement is that the highest-order d bits in the binary representation of z(x − y) mod 2w are either all 0’s or all 1’s. That is, z(x − y) mod 2w = (0, . . . , 0, ?, . . . , ?)2 (5.2) | {z } | {z } d w−d when zx mod 2w > zy mod 2w or z(x − y) mod 2w = (1, . . . , 1, ?, . . . , ?)2 . (5.3) | {z } | {z } d w−d when zx mod 2w < zy mod 2w . Therefore, we only have to bound the probability that z(x − y) mod 2w looks like (5.2) or (5.3). Let q be the unique odd integer such that (x − y) mod 2w = q2r for some integer r ≥ 0. By Lemma 5.3, the binary representation of zq mod 2w has w − 1 random bits, followed by a 1: zq mod 2w = (bw−1 , . . . , b1 , 1)2 | {z } w−1 Therefore, the binary representation of z(x − y) mod 2w = zq2r mod 2w has w − r − 1 random bits, followed by a 1, followed by r 0’s: z(x − y) mod 2w = zq2r mod 2w = (bw−r−1 , . . . , b1 , 1, 0, 0, . . . , 0)2 | {z } | {z } w−r−1 r We can now finish the proof: If r > w − d, then the d higher order bits of z(x − y) mod 2w contain both 0’s and 1’s, so the probability that z(x − y) mod 2w looks like (5.2) or (5.3) is 0. If r = w − d, then the probability of looking like (5.2) is 0, but the probability of looking like (5.3) is 1/2d−1 = 2/2d (since we must have b1 , . . . , bd−1 = 1, . . . , 1). If r < w − d then we must have bw−r−1 , . . . , bw−r−d = 0, . . . , 0 or bw−r−1 , . . . , bw−r−d = 1, . . . , 1. The probability of each of these cases is 1/2d and they are mutually exclusive, so the probability of either of these cases is 2/2d . This completes the proof. 84 5. Hash Tables 5.2. LinearHashTable: Linear Probing 5.1.2 Summary The following theorem summarizes the performance of the ChainedHashTable data struc- ture: Theorem 5.1. A ChainedHashTable implements the USet interface. Ignoring the cost of calls to grow(), a ChainedHashTable supports the operations add(x), remove(x), and find(x) in O(1) expected time per operation. Furthermore, beginning with an empty ChainedHashTable, any sequence of m add(x) and remove(x) operations results in a total of O(m) time spent during all calls to grow(). 5.2 LinearHashTable: Linear Probing The ChainedHashTable data structure uses an array of lists, where the ith list stores all elements x such that hash(x) = i. An alternative, called open addressing is to store the elements directly in an array, t, with each array location in t storing at most one value. This is the approach taken by the LinearHashTable described in this section. In some places, this data structure is described as open addressing with linear probing. The main idea behind a LinearHashTable is that we would, ideally, like to store the element x with hash value i = hash(x) in the table location t[i]. If we can’t do this (because some element is already stored there) then we try to store it at location t[(i + 1) mod t.length]; if that’s not possible, then we try t[(i + 2) mod t.length], and so on, until we find a place for x. There are three types of entries stored in t: 1. data values: actual values in the USet that we are representing; 2. null values: at array locations where no data has ever been stored; and 3. del values: at array locations where data was once stored but that has since been deleted. In addition to the counter, n, that keeps track of the number of elements in the LinearHashTable, a counter, q, keeps track of the number of elements of Types 1 and 3. That is, q is equal to n plus the number of del values in t. To make this work efficiently, we need t to be considerably larger than q, so that there are lots of null values in t. The operations on a LinearHashTable therefore maintain the invariant that t.length ≥ 2q. To summarize, a LinearHashTable contains an array, t, that stores data elements, and integers n and q that keep track of the number of data elements and non-null values 85 5. Hash Tables 5.2. LinearHashTable: Linear Probing of t, respectively. Because many hash functions only work for table sizes that are a power of 2, we also keep an integer d and maintain the invariant that t.length = 2d . LinearHashTable array<T> t; int n; // number of values in T int q; // number of non-null entries in T int d; // t.length = 2^d The find(x) operation in a LinearHashTable is simple. We start at array entry t[i] where i = hash(x) and search entries t[i], t[(i+1) mod t.length], t[(i+2) mod t.length], and so on, until we find an index i0 such that, either, t[i0 ] = x, or t[i0 ] = null. In the former case we return t[i0 ]. In the latter case, we conclude that x is not contained in the hash table and return null. LinearHashTable T find(T x) { int i = hash(x); while (t[i] != null) { if (t[i] != del && t[i] == x) return t[i]; i = (i == t.length-1) ? 0 : i + 1; // increment i (mod t.length) } return null; } The add(x) operation is also fairly easy to implement. After checking that x is not already stored in the table (using find(x)), we search t[i], t[(i + 1) mod t.length], t[(i + 2) mod t.length], and so on, until we find a null or del and store x at that location, increment n, and q, if appropriate.: LinearHashTable bool add(T x) { if (find(x) != null) return false; if (2*(q+1) > t.length) resize(); // max 50% occupancy int i = hash(x); while (t[i] != null && t[i] != del) i = (i == t.length-1) ? 0 : i + 1; // increment i (mod t.length) if (t[i] == null) q++; n++; t[i] = x; return true; } By now, the implementation of the remove(x) operation should be obvious. we search t[i], t[(i + 1) mod t.length], t[(i + 2) mod t.length], and so on until we find an 86 5. Hash Tables 5.2. LinearHashTable: Linear Probing index i0 such that t[i0 ] = x or t[i0 ] = null. In the former case, we set t[i0 ] = del and return true. In the latter case we conclude that x was not stored in the table (and therefore cannot be deleted) and return false. LinearHashTable T remove(T x) { int i = hash(x); while (t[i] != null) { T y = t[i]; if (y != del && x == y) { t[i] = del; n--; if (8*n < t.length) resize(); // min 12.5% occupancy return y; } i = (i == t.length-1) ? 0 : i + 1; // increment i (mod t.length) } return null; } The correctness of the find(x), add(x), and remove(x) methods is easy to verify, though it relies on the use of del values. Notice that none of these operations ever sets a non-null entry to null. Therefore, when we reach an index i0 such that t[i0 ] = null, this is a proof that the element, x, that we are searching for is not stored in the table; t[i0 ] has always been null, so there is no reason that a previous add(x) operation would have proceeded beyond index i0 . The resize() method is called by add(x) when the number of non-null entries exceeds n/2 or by remove(x) when the number of data entries is less than t.length/8. The resize() method works like the resize() methods in other array-based data structures. We find the smallest non-negative integer d such that 2d ≥ 3n. We reallocate the array t so that it has size 2d and then we insert all the elements in the old version of t into the newly-resized copy of t. While doing this we reset q equal to n since the newly-allocated t has no del values. LinearHashTable void resize() { d = 1; while ((1<<d) < 3*n) d++; array<T> tnew(1<<d, null); q = n; // insert everything in told for (int k = 0; k < t.length; k++) { if (t[k] != null && t[k] != del) { 87 5. Hash Tables 5.2. LinearHashTable: Linear Probing int i = hash(t[k]); while (tnew[i] != null) i = (i == tnew.length-1) ? 0 : i + 1; tnew[i] = t[k]; } } t = tnew; } 5.2.1 Analysis of Linear Probing Notice that each operation, add(x), remove(x), or find(x), finishes as soon as (or before) it discovers the first null entry in t. The intuition behind the analysis of linear probing is that, since at least half the elements in t are equal to null, an operation should not take long to complete because it will very quickly come across a null entry. We shouldn’t rely too heavily on this intuition though, because it would lead us to (the incorrect) conclusion that the expected number of locations in t examined by an operation is at most 2. For the rest of this section, we will assume that all hash values are independently and uniformly distributed in {0, . . . , t.length−1}. This is not a realistic assumption, but it will make it possible for us to analyze linear probing. Later in this section we will describe a method, called tabulation hashing, that produces a hash function that is “good enough” for linear probing. We will also assume that all indices into the positions of t are taken modulo t.length, so that t[i] is really a shorthand for t[i mod t.length]. A run of length k that starts at i occurs when t[i], t[i + 1], . . . , t[i + k − 1] are all non-null and t[i − 1] = t[i + k] = null. The number of non-null elements of t is exactly q and the add(x) method ensures that, at all times, q ≤ t.length/2. There are q elements x1 , . . . , xq that have been inserted into t since the last rebuild() operation. By our assumption, each of these has a hash value, hash(xj ), that is uniform and independent of the rest. With this setup, we can prove the main lemma required to analyze linear probing. Lemma 5.4. For any i ∈ {0, . . . , t.length−1}, the probability that a run of length k starts at i is O(ck ) for some constant 0 < c < 1. Proof. If a run of length k starts at i, then there are exactly k elements xj such that hash(xj ) ∈ {i, . . . , i + k − 1}. The probability that this occurs is exactly k q k t.length − k q−k pk = , k t.length t.length 88 5. Hash Tables 5.2. LinearHashTable: Linear Probing since, for each choice of k elements, these k elements must hash to one of the k locations and the remaining q − k elements must hash to the other t.length − k table locations.1 In the following derivation we will cheat a little and replace r! with (r/e)r . Stirling’s √ Approximation (Section 1.2.2) shows that this is only a factor of O( r) from the truth. This is just done to make the derivation simpler; Exercise 5.2 asks the reader to redo the calculation more rigorously using Stirling’s Approximation in its entirety. The value of pk is maximized when t.length is minimum, and the data structure maintains the invariant that t.length ≥ 2q, so k q k 2q − k q−k pk ≤ k 2q 2q k q! k 2q − k q−k = (q − k)!k! 2q 2q k qq k 2q − k q−k ≈ [Stirling’s approximation] (q − k)q−k k k 2q 2q k k q q q−k k 2q − k q−k = (q − k)q−k k k 2q 2q k q−k qk q(2q − k) = 2qk 2q(q − k) k 1 (2q − k) q−k = 2 2(q − k) k q−k 1 k = 1+ 2 2(q − k) √ k e = . 2 (In the last step, we use the inequality (1 + 1/x)x ≤ e, which holds for all x > 0.) Since √ e/2 < 0.824360636 < 1, this completes the proof. Using Lemma 5.4 to prove upper-bounds on the expected running time of find(x), add(x), and remove(x) is now fairly straight-forward. Consider the simplest case, where we execute find(x) for some value x that has never been stored in the LinearHashTable. In this case, i = hash(x) is a random value in {0, . . . , t.length − 1} independent of the contents of t. If i is part of a run of length k then the time it takes to execute the find(x) 1 Note that pk is greater than the probability that a run of length k starts at i, since the definition of pk does not include the requirement t[i − 1] = t[i + k] = null. 89 5. Hash Tables 5.2. LinearHashTable: Linear Probing operation is at most O(1 + k). Thus, the expected running time can be upper-bounded by t.length ∞ ! 1 X X O 1+ k Pr{i is part of a run of length k} . t.length i=1 k=0 Note that each run of length k contributes to the inner sum k times for a total contribution of k 2 , so the above sum can be rewritten as t.length ∞ ! 1 X X 2 O 1+ k Pr{i starts a run of length k} t.length i=1 k=0 t.length ∞ ! 1 X X ≤O 1+ k 2 pk t.length i=1 k=0 ∞ ! X =O 1+ k 2 pk k=0 ∞ ! X =O 1+ k 2 · O(ck ) k=0 = O(1) . P∞ 2 · O(ck ) The last step in this derivation comes from the fact that k=0 k is an exponentially decreasing series.2 Therefore, we conclude that the expected running time of the find(x) operation for a value x that is not contained in a LinearHashTable is O(1). If we ignore the cost of the resize() operation, the above analysis gives us all we need to analyze the cost of operations on a LinearHashTable. First of all, the analysis of find(x) given above applies to the add(x) operation when x is not contained in the table. To analyze the find(x) operation when x is contained in the table, we need only note that this is the same as the cost of the add(x) operation that previously added x to the table. Finally, the cost of a remove(x) operation is the same as the cost of a find(x) operation. In summary, if we ignore the cost of calls to resize(), all operations on a LinearHashTable run in O(1) expected time. Accounting for the cost of resize can be done using the same type of amortized analysis performed for the ArrayStack data structure in Section 2.1. 5.2.2 Summary The following theorem summarizes the performance of the LinearHashTable data structure: 2 In the terminology of many calculus texts, this sum passes the ratio test: There exists a positive integer (k+1)2 ck+1 k0 such that, for all k ≥ k0 , k2 ck < 1. 90 5. Hash Tables 5.2. LinearHashTable: Linear Probing Theorem 5.2. A LinearHashTable implements the USet interface. Ignoring the cost of calls to resize(), a LinearHashTable supports the operations add(x), remove(x), and find(x) in O(1) expected time per operation. Furthermore, beginning with an empty LinearHashTable, any sequence of m add(x) and remove(x) operations results in a total of O(m) time spent during all calls to resize(). 5.2.3 Tabulation Hashing While analyzing the LinearHashTable structure, we made a very strong assumption: That for any set of elements, {x1 , . . . , xn }, the hash values hash(x1 ), . . . , hash(xn ) are indepen- dently and uniformly distributed over {0, . . . , t.length − 1}. One way to imagine getting this is to have a giant array, tab, of length 2w , where each entry is a random w-bit integer, independent of all the other entries. In this way, we could implement hash(x) by extracting a d-bit integer from tab[x.hashCode()]: LinearHashTable int idealHash(T x) { return tab[hashCode(x) >> w-d]; } Unfortunately, storing an array of size 2w is prohibitive in terms of memory usage. The approach used by tabulation hashing is to, instead, store w/r arrays each of length 2r . All the entries in these arrays are indepent w-bit integers. To obtain the value of hash(x) we split x.hashCode() up into w/r r-bit integers and use these as indices into these arrays. We then combine all these values with the bitwise exclusive-or operator to obtain hash(x). The following code shows how this works when w = 32 and r = 4: LinearHashTable int hash(T x) { unsigned h = hashCode(x); return (tab[0][h&0xff] ^ tab[1][(h>>8)&0xff] ^ tab[2][(h>>16)&0xff] ^ tab[3][(h>>24)&0xff]) >> (w-d); } In this case, tab is a 2-dimensional array with 4 columns and 232/4 = 256 rows. One can easily verify that, for any x, hash(x) is uniformly distributed over {0, . . . , 2d − 1}. With a little work, one can even verify that any pair of values have independent hash 91 5. Hash Tables 5.3. Hash Codes values. This implies tabulation hashing could be used in place of multiplicative hashing for the ChainedHashTable implementation. However, it is not true that any set of n distinct values gives a set of n independent hash values. Nevertheless, when tabulation hashing is used, the bound of Theorem 5.2 still holds. References for this are provided at the end of this chapter. 5.3 Hash Codes The hash tables discussed in the previous section are used to associate data with integer keys consisting of w bits. In many cases, we have keys that are not integers. They may be strings, objects, arrays, or other compound structures. To use hash tables for these types of data, we must map these data types to w-bit hash codes. Hash code mappings should have the following properties: 1. If x and y are equal, then x.hashCode() and y.hashCode() are equal. 2. If x and y are not equal, then the probability that x.hashCode() = y.hashCode() should be small (close to 1/2w ). The first property ensures that if we store x in a hash table and later look up a value y equal to x, then we will find x—as we should. The second property minimizes the loss from converting our objects to integers. It ensures that unequal objects usually have different hash codes and so are likely to be stored at different locations in our hash table. 5.3.1 Hash Codes for Primitive Data Types Small primitive data types like char, byte, int, and float are usually easy to find hash codes for. These data types always have a binary representation and this binary representa- tion usually consists of w or fewer bits. (For example, in C++ char is typically an 8-bit type and float is a 32-bit type.) In these cases, we just treat these bits as the representation of an integer in the range {0, . . . , 2w − 1}. If two values are different, they get different hash codes. If they are the same, they get the same hash code. A few primitive data types are made up of more than w bits, usually cw bits for some constant integer c. (Java’s long and double types are examples of this with c = 2.) These data types can be treated as compound objects made of c parts, as described in the next section. 92 5. Hash Tables 5.3. Hash Codes 5.3.2 Hash Codes for Compound Objects For a compound object, we want to create a hash code by combining the individual hash codes of the object’s constituent parts. This is not as easy as it sounds. Although one can find many hacks for this (for example, combining the hash codes with bitwise exclusive- or operations), many of these hacks turn out to be easy to foil (see Exercises 5.4–5.6). However, if one is willing to do arithmetic with 2w bits of precision, then there are simple and robust methods available. Suppose we have an object made up of several parts P0 , . . . , Pr−1 whose hash codes are x0 , . . . , xr−1 . Then we can choose mutually independent random w-bit integers z0 , . . . , zr−1 and a random 2w-bit odd integer z and compute a hash code for our object with ! ! r−1 X h(x0 , . . . , xr−1 ) = z zi xi mod 22w div 2w . i=0 Note that this hash code has a final step (multiplying by z and dividing by 2w ) that uses the multiplicative hash function from Section 5.1.1 to take the 2w-bit intermediate result and reduce it to a w-bit final result. Here is an example of this method applied to a simple compound object with 3 parts x0, x1, and x2: Point3D unsigned hashCode() { long long z[] = {0x2058cc50L, 0xcb19137eL, 0x2cb6b6fdL}; // random long zz = 0xbea0107e5067d19dL; // random long h0 = ods::hashCode(x0); long h1 = ods::hashCode(x1); long h2 = ods::hashCode(x2); return (int)(((z[0]*h0 + z[1]*h1 + z[2]*h2)*zz) >> 32); } The following theorem shows that, in addition to being straightforward to implement, this method is provably good: Theorem 5.3. Let x0 , . . . , xr−1 and y0 , . . . , yr−1 each be sequences of w bit integers in {0, . . . , 2w − 1} and assume xi 6= yi for at least one index i ∈ {0, . . . , r − 1}. Then Pr{h(x0 , . . . , xr−1 ) = h(y0 , . . . , yr−1 )} ≤ 3/2w } . Proof. We will first ignore the final multiplicative hashing step and see how that step contributes later. Define: r−1 X h0 (x0 , . . . , xr−1 ) = zj xj mod 22w . j=0 93 5. Hash Tables 5.3. Hash Codes Suppose that h0 (x0 , . . . , xr−1 ) = h0 (y0 , . . . , yr−1 ). We can rewrite this as: zi (xi − yi ) mod 22w = t (5.4) where i−1 X r−1 X t= zj (yj − xj ) + zj (yj − xj ) mod 22w j=0 j=i+1 If we assume, without loss of generality that xi > yi , then (5.4) becomes zi (xi − yi ) = t , (5.5) since each of zi and (xi − yi ) is at most 2w − 1, so their product is at most 22w − 2w+1 + 1 < 22w − 1. By assumption, xi − yi 6= 0, so (5.5) has at most one solution in zi . Therefore, since zi and t are independent (z0 , . . . , zr−1 are mutually independent), the probability that we select zi so that h0 (x0 , . . . , xr−1 ) = h0 (y0 , . . . , yr−1 ) is at most 1/2w . The final step of the hash function is to apply multiplicative hashing to reduce our 2w-bit intermediate result h0 (x0 , . . . , xr−1 ) to a w-bit final result h(x0 , . . . , xr−1 ). By Theo- rem 5.3, if h0 (x0 , . . . , xr−1 ) 6= h0 (y0 , . . . , yr−1 ), then Pr{h(x0 , . . . , xr−1 ) = h(y0 , . . . , yr−1 )} ≤ 2/2w . To summarize, ( ) h(x0 , . . . , xr−1 ) Pr = h(y0 , . . . , yr−1 ) 0 0 h (x0 , . . . , xr−1 ) = h (y0 , . . . , yr−1 ) or = Pr h0 (x0 , . . . , xr−1 ) 6= h0 (y0 , . . . , yr−1 ) and zh0 (x0 , . . . , xr−1 ) div 2w = zh0 (y0 , . . . , yr−1 ) div 2w ≤ 1/2w + 2/2w = 3/2w . 5.3.3 Hash Codes for Arrays and Strings The method from the previous section works well for objects that have a fixed, constant, number of components. However, it breaks down when we want to use it with objects that have a variable number of components since it requires a random w-bit integer zi for each component. We could use a pseudorandom sequence to generate as many zi ’s as we need, but then the zi ’s are not mutually independent, and it becomes difficult to prove that the pseudorandom numbers don’t interact badly with the hash function we are using. In particular, the values of t and zi in the proof of Theorem 5.3 are no longer independent. 94 5. Hash Tables 5.3. Hash Codes A more rigorous approach is to base our hash codes on polynomials over prime fields. This method is based on the following theorem, which says that polynomials over prime fields behave pretty-much like usual polynomials: Theorem 5.4. Let p be a prime number, and let f (z) = x0 z0 +x1 z1 +· · ·+xr−1 zr−1 be a non- trivial polynomial with coefficients xi ∈ {0, . . . , p − 1}. Then the equation f (z) mod p = 0 has at most r − 1 solutions for z ∈ {0, . . . , p − 1}. To use Theorem 5.4, we hash a sequence of integers x0 , . . . , xr−1 with each xi ∈ {0, . . . , p − 2} using a random integer z ∈ {0, . . . , p − 1} via the formula h(x0 , . . . , xr−1 ) = x0 z0 + · · · + xr−1 zr−1 + (p − 1)zr mod p . Note the extra (p − 1)zr term at the end of the formula. It helps to think of (p − 1) as the last element, xr , in the sequence x0 , . . . , xr . Note that this element differs from every other element in the sequence (each of which is in the set {0, . . . , p − 2}). We can think of p − 1 as an end-of-sequence marker. The following theorem, which considers the case of two sequences of the same length, shows that this hash function gives a good return for the small amount of randomization needed to choose z: Theorem 5.5. Let p > 2w +1 be a prime, let x0 , . . . , xr−1 and y0 , . . . , yr−1 each be sequences of w-bit integers in {0, . . . , 2w −1}, and assume xi 6= yi for at least one index i ∈ {0, . . . , r−1}. Then Pr{h(x0 , . . . , xr−1 ) = h(y0 , . . . , yr−1 )} ≤ (r − 1)/p} . Proof. The equation h(x0 , . . . , xr−1 ) = h(y0 , . . . , yr−1 ) can be rewritten as (x0 − y0 )z0 + · · · + (xr−1 − yr−1 )zr−1 mod p = 0. (5.6) Since xi 6= yi , this polynomial is non-trivial. Therefore, by Theorem 5.4, it has at most r − 1 solutions in z. The probability that we pick z to be one of these solutions is therefore at most (r − 1)/p. Note that this hash function also deals with the case in which two sequences have different lengths, even when one of the sequences is a prefix of the other. This is because this function effectively hashes the infinite sequence x0 , . . . , xr−1 , p − 1, 0, 0, . . . . 95 5. Hash Tables 5.3. Hash Codes This guarantees that if we have two sequences of length r and r0 with r > r0 , then these two sequences differ at index i = r. In this case, (5.6) becomes 0 ! X−1 i=r i=r−1 X i r0 i r (xi − yi )z + (xr0 − p + 1)z + xi z + (p − 1)z mod p = 0 , i=0 i=r0 +1 which, by Theorem 5.4, has at most r solutions in z. This combined with Theorem 5.5 suffice to prove the following more general theorem: Theorem 5.6. Let p > 2w + 1 be a prime, let x0 , . . . , xr−1 and y0 , . . . , yr0 −1 be distinct sequences of w-bit integers in {0, . . . , 2w − 1}. Then Pr{h(x0 , . . . , xr−1 ) = h(y0 , . . . , yr−1 )} ≤ max{r, r0 }/p} . The following example code shows how this hash function is applied to an object that contains an array, x, of values: GeomVector unsigned hashCode() { long p = (1L<<32)-5; // prime: 2^32 - 5 long z = 0x64b6055aL; // 32 bits from random.org int z2 = 0x5067d19d; // random odd 32 bit number long s = 0; long zi = 1; for (int i = 0; i < x.length; i++) { long long xi = (ods::hashCode(x[i]) * z2) >> 1; // reduce to 31 bits s = (s + zi * xi) % p; zi = (zi * z) % p; } s = (s + zi * (p-1)) % p; return (int)s; } The above code sacrifices some collision probability for implementation convenience. In particular, it applies the multiplicative hash function from Section 5.1.1, with d = 31 to reduce x[i].hashCode() to a 31-bit value. This is so that the additions and multiplications that are done modulo the prime p = 232 − 5 can be carried out using unsigned 63-bit arithmetic. This means that the probability of two different sequences, the longer of which has length r, having the same hash code is at most 2/231 + r/(232 − 5) rather than the r/(232 − 5) specified in Theorem 5.6. 96 5. Hash Tables 5.4. Discussion and Exercises 5.4 Discussion and Exercises Hash tables and hash codes are an enormous and active area of research that is just touched upon in this chapter. The online Bibliography on Hashing [7] contains nearly 2000 entries. A variety of different hash table implementations exist. The one described in Sec- tion 5.1 is known as hashing with chaining (each array entry contains a chain (List) of elements). Hashing with chaining dates back to an internal IBM memorandum authored by H. P. Luhn and dated January 1953. This memorandum also seems to be one of the earliest references to linked lists. An alternative to hashing with chaining is that used by open addressing schemes, where all data is stored directly in an array. These schemes include the LinearHashTable structure of Section 5.2. This idea was also proposed, independently, by a group at IBM in the 1950s. Open addressing schemes must deal with the problem of collision resolution: the case where two values hash to the same array location. Different strategies exist for collision resolution and these provide different performance guarantees and often require more sophisticated hash functions than the ones described here. Yet another category of hash table implementations are the so-called perfect hashing methods. These are methods in which find(x) operations take O(1) time in the worst-case. For static data sets, this can be accomplished by finding perfect hash functions for the data; these are functions that map each piece of data to a unique array location. For data that changes over time, perfect hashing methods include FKS two-level hash tables [25, 19] and cuckoo hashing [46]. The hash functions presented in this chapter are probably among the most practical currently known methods that can be proven to work well for any set of data. Other provably good methods date back to the pioneering work of Carter and Wegman who introduced the notion of universal hashing and described several hash functions for different scenarios [11]. Tabulation hashing, described in Section 5.2.3, is due to Carter and Wegman [11], but its analysis, when applied to linear probing (and several other hash table schemes) is due to Pǎtraşcu and Thorup [51]. The idea of multiplicative hashing is very old and seems to be part of the hashing folklore [41, Section 6.4]. However, the idea of choosing the multiplier z to be a random odd number, and the analysis in Section 5.1.1 is due to Dietzfelbinger et al. [18]. This version of multiplicative hashing is one of the simplest, but its collision probability of 2/2d is a factor of 2 larger than what one could expect with a random function from 2w → 2d . The 97 5. Hash Tables 5.4. Discussion and Exercises multiply-add hashing method uses the function h(x) = ((zx + b) mod 22w ) div 22w−d where z and b are each randomly chosen from {0, . . . , 22w − 1}. Multiply-add hashing has a collision probability of only 1/2d [16], but requires 2w-bit precision arithmetic. There are a number of methods of obtaining hash codes from fixed-length sequences of w-bit integers. One particularly fast method [8] is the function r/2−1 X h(x0 , . . . , xr−1 ) = ((x2i + a2i ) mod 2w )((x2i+1 + a2i+1 ) mod 2w ) mod 22w i=0 where r is even and a0 , . . . , ar−1 are randomly chosen from {0, . . . , 2w }. This yields a 2w-bit hash code that has collision probability 1/2w . This can be reduced to a w-bit hash code using multiplicative (or multiply-add) hashing. This method is fast because it requires only r/2 2w-bit multiplications whereas the method described in Section 5.3.2 requires r multiplications. (The mod operations occur implicitly by using w and 2w-bit arithmetic for the additions and multiplications, respectively.) The method from Section 5.3.3 of using polynomials over prime fields to hash variable-length arrays and strings is due to Dietzfelbinger et al. [17]. It is, unfortunately, not very fast. This is due to its use of the mod operator which relies on a costly machine instruction. Some variants of this method choose the prime p to be one of the form 2w − 1, in which case the mod operator can be replaced with addition (+) and bitwise-and (&) operations [40, Section 3.6]. Another option is to apply one of the fast methods for fixed- length strings to blocks of length c for some constant c > 1 and then apply the prime field method to the resulting sequence of dr/ce hash codes. Exercise 5.1. Prove that the bound 2/2d in Lemma 5.1 is the best possible by showing that, if x = 2w−d−2 and y = 3x, then Pr{hash(x) = hash(y)} = 2/2d . (Hint look at the binary representations of zx and z3x and use the fact that z3x = zx+2zx.) Exercise 5.2. Reprove Lemma 5.4 using the full version of Stirling’s Approximation given in Section 1.2.2. Exercise 5.3. Consider the following the simplified version of the code for adding an element x to a LinearHashTable. This code simply stores x in the first null array entry it finds. Explain why this could be very slow by giving an example of a sequence of O(n) add(x), remove(x), and find(x) operations that would take on the order of n2 time to execute. 98 5. Hash Tables 5.4. Discussion and Exercises LinearHashTable bool addSlow(T x) { if (2*(q+1) > t.length) resize(); // max 50% occupancy int i = hash(x); while (t[i] != null) { if (t[i] != del && x.equals(t[i])) return false; i = (i == t.length-1) ? 0 : i + 1; // increment i (mod t.length) } t[i] = x; n++; q++; return true; } Exercise 5.4. Suppose you have an object made up of two w-bit integers x and y. Show why x ⊕ y does not make a good hash code for your object. Give an example of a large set of objects that would all have hash code 0. Exercise 5.5. Suppose you have an object made up of two w-bit integers x and y. Show why x + y does not make a good hash code for your object. Give an example of a large set of objects that would all have the same hash code. Exercise 5.6. Suppose you have an object made up of two w-bit integers x and y. Suppose that the hash code for your object is defined by some deterministic function h(x, y). Prove that there exists a large set of objects that have the same hash code. Exercise 5.7. Let p = 2w − 1 for some positive integer w. Explain why, for a positive integer x (x mod 2w ) + (x div 2w ) ≡ x mod (2w − 1) . (This gives an algorithm for computing x mod (2w − 1) by repeatedly setting x = x&((1 << w) − 1) + x >>> w until x ≤ 2w − 1.) 99 5. Hash Tables 5.4. Discussion and Exercises 100 Chapter 6 Binary Trees This chapter introduces one of the most fundamental structures in computer science: binary trees. There are lots of ways of defining binary trees. Mathematically, a binary tree is a connected undirected finite graph with no cycles, and no vertex of degree greater than three. For most computer science applications, binary trees are rooted : A special node, r, of degree at most two is called the root of the tree. For every node, u 6= r, the second node on the path from u to r is called the parent of u. Each of the other nodes adjacent to u is called a child of u. Most of the binary trees we are interested in are ordered, so we distinguish between the left child and right child of u. In illustrations, binary trees are usually drawn from the root downward, with the root at the top of the drawing and the left and right children respectively given by left and right positions in the drawing (Figure 6.1). A binary tree with nine nodes is drawn this way in Figure 6.2.a. Binary trees are so important that a terminology has developed around them: The depth of a node, u, in a binary tree is the length of the path from u to the root of the tree. u.parent u u.left u.right Figure 6.1: The parent, left child, and right child of the node u in a BinaryTree. 101 6. Binary Trees 6.1. BinaryTree: A Basic Binary Tree r r (a) (b) Figure 6.2: A binary tree with (a) nine real nodes and (b) ten external nodes. If a node, w, is on the path from u to r then w is called an ancestor of u and u a descendant of w. The subtree of a node, u, is the binary tree that is rooted at u and contains all of u’s descendants. The height of a node, u, is the length of the longest path from u to one of its descendants. The height of a tree is the height of its root. A node, u, is a leaf if it has no children. We sometimes think of the tree as being augmented with external nodes. Any node that does not have a left child has an external node as its left child and any node that does not have a right child has an external node as its right child (see Figure 6.2.b). It is easy to verify, by induction, that a binary tree having n ≥ 1 real nodes has n + 1 external nodes. 6.1 BinaryTree: A Basic Binary Tree The simplest way to represent a node, u, in a binary tree is to store the (at most three) neighbours of u explicitly: BinaryTree class BTNode { N *left; N *right; N *parent; BTNode() { left = right = parent = NULL; } }; When one of these three neighbours is not present, we set it to nil. In this way, external nodes in the tree as well as the parent of the root correspond to the value nil. The binary tree itself can then be represented by a pointer to its root node, r: 102 6. Binary Trees 6.1. BinaryTree: A Basic Binary Tree BinaryTree Node *r; // root node We can compute the depth of a node, u, in a binary tree by counting the number of steps on the path from u to the root: BinaryTree int depth(Node *u) { int d = 0; while (u != r) { u = u->parent; d++; } return d; } 6.1.1 Recursive Algorithms It is very easy to compute facts about binary trees using recursive algorithms. For example, to compute the size of (number of nodes in) a binary tree rooted at node u, we recursively compute the sizes of the two subtrees rooted at the children of u, sum these sizes, and add one: BinaryTree int size(Node *u) { if (u == nil) return 0; return 1 + size(u->left) + size(u->right); } To compute the height of a node u we can compute the height of u’s two subtrees, take the maximum, and add one: BinaryTree int height(Node *u) { if (u == nil) return -1; return 1 + max(height(u->left), height(u->right)); } 6.1.2 Traversing Binary Trees The two algorithms from the previous section use recursion to visit all the nodes in a binary tree. Each of them visits the nodes of the binary tree in the same order as the following code: 103 6. Binary Trees 6.1. BinaryTree: A Basic Binary Tree BinaryTree void traverse(Node *u) { if (u == nil) return; traverse(u->left); traverse(u->right); } Using recursion this way produces very short, simple code, but can be problematic. The maximum depth of the recursion is given by the maximum depth of a node in the binary tree, i.e., the tree’s height. If the height of the tree is very large, then this could very well use more stack space than is available, causing a crash. Luckily, traversing a binary tree can be done without recursion. This is done using an algorithm that uses where it came from to decide where it will go next. See Figure 6.3. If we arrive at a node u from u.parent, then the next thing to do is to visit u.left. If we arrive at u from u.left, then the next thing to do is to visit u.right. If we arrive at u from u.right, then we are done visiting u’s subtree, so we return to u.parent. The following code implements this idea, with code included for handling the cases where any of u.left, u.right, or u.parent is nil: BinaryTree void traverse2() { Node *u = r, *prev = nil, *next; while (u != nil) { if (prev == u->parent) { if (u->left != nil) next = u->left; else if (u->right != nil) next = u->right; else next = u->parent; } else if (prev == u->left) { if (u->right != nil) next = u->right; else next = u->parent; } else { next = u->parent; } prev = u; u = next; } } The same things that can be computed with recursive algorithms can also be done this way. For example, to compute the size of the tree we keep a counter, n, and increment n whenever visiting a node for the first time: 104 6. Binary Trees 6.1. BinaryTree: A Basic Binary Tree r u.parent u u.left u.right Figure 6.3: The three cases that occur at node u when traversing a binary tree non- recursively, and the resulting traversal of the tree. BinaryTree int size2() { Node *u = r, *prev = nil, *next; int n = 0; while (u != nil) { if (prev == u->parent) { n++; if (u->left != nil) next = u->left; else if (u->right != nil) next = u->right; else next = u->parent; } else if (prev == u->left) { if (u->right != nil) next = u->right; else next = u->parent; } else { next = u->parent; } prev = u; u = next; } return n; } In some implementations of binary trees, the parent field is not used. When this is the case, a non-recursive implementation is still possible, but the implementation has to use a List (or Stack) to keep track of the path from the current node to the root. A special kind of traversal that does not fit the pattern of the above functions is the breadth-first traversal. In a breadth-first traversal, the nodes are visited level-by-level starting at the root and working our way down, visiting the nodes at each level from left 105 6. Binary Trees 6.2. BinarySearchTree: An Unbalanced Binary Search Tree r Figure 6.4: During a breadth-first traversal, the nodes of a binary tree are visited level-by- level, and left-to-right within each level. to right. This is similar to the way we would read a page of English text. (See Figure 6.4.) This is implemented using a queue, q, that initially contains only the root, r. At each step, we extract the next node, u, from q, process u and add u.left and u.right (if they are non-nil) to q: BinaryTree void bfTraverse() { ArrayDeque<Node*> q; if (r != nil) q.add(q.size(),r); while (q.size() > 0) { Node *u = q.remove(q.size()-1); if (u->left != nil) q.add(q.size(),u->left); if (u->right != nil) q.add(q.size(),u->right); } } 6.2 BinarySearchTree: An Unbalanced Binary Search Tree A BinarySearchTree is a special kind of binary tree in which each node, u, also stores a data value, u.x, from some total order. The data values in a binary search tree obey the binary search tree property: For a node, u, every data value stored in the subtree rooted at u.left is less than u.x and every data value stored in the subtree rooted at u.right is greater than u.x. An example of a BinarySearchTree is shown in Figure 6.5. 6.2.1 Searching The binary search tree property is extremely useful because it allows us to quickly locate a value, x, in a binary search tree. To do this we start searching for x at the root, r. When examining a node, u, there are three cases: 106 6. Binary Trees 6.2. BinarySearchTree: An Unbalanced Binary Search Tree 7 3 11 1 5 9 13 4 6 8 12 14 Figure 6.5: A binary search tree. 1. If x < u.x then the search proceeds to u.left; 2. If x > u.x then the search proceeds to u.right; 3. If x = u.x then we have found the node u containing x. The search terminates when Case 3 occurs or when u = nil. In the former case, we found x. In the latter case, we conclude that x is not in the binary search tree. BinarySearchTree T findEQ(T x) { Node *w = r; while (w != nil) { int comp = compare(x, w->x); if (comp < 0) { w = w->left; } else if (comp > 0) { w = w->right; } else { return w->x; } } return null; } Two examples of searches in a binary search tree are shown in Figure 6.6. As the second example shows, even if we don’t find x in the tree, we still gain some valuable information. If we look at the last node, u, at which Case 1 occurred, we see that u.x is the smallest value in the tree that is greater than x. Similarly, the last node at which Case 2 107 6. Binary Trees 6.2. BinarySearchTree: An Unbalanced Binary Search Tree 7 7 3 11 3 11 1 5 9 13 1 5 9 13 4 6 8 12 14 4 6 8 12 14 (a) (b) Figure 6.6: An example of (a) a successful search (for 6) and (b) an unsuccessful search (for 10) in a binary search tree. occurred contains the largest value in the tree that is less than x. Therefore, by keeping track of the last node, z, at which Case 1 occurs, a BinarySearchTree can implement the find(x) operation that returns the smallest value stored in the tree that is greater than or equal to x: BinarySearchTree T find(T x) { Node *w = r, *z = nil; while (w != nil) { int comp = compare(x, w->x); if (comp < 0) { z = w; w = w->left; } else if (comp > 0) { w = w->right; } else { return w->x; } } return z == nil ? null : z->x; } 6.2.2 Inserting To add a new value, x, to a BinarySearchTree, we first search for x. If we find it, then there is no need to insert it. Otherwise, we store x at a leaf child of the last node, p, encountered during the search for x. Whether the new node is the left or right child of p depends on the 108 6. Binary Trees 6.2. BinarySearchTree: An Unbalanced Binary Search Tree result of comparing x and p.x. BinarySearchTree bool add(T x) { Node *p = findLast(x); Node *u = new Node; u->x = x; return addChild(p, u); } BinarySearchTree Node* findLast(T x) { Node *w = r, *prev = nil; while (w != nil) { prev = w; int comp = compare(x, w->x); if (comp < 0) { w = w->left; } else if (comp > 0) { w = w->right; } else { return w; } } return prev; } BinarySearchTree bool addChild(Node *p, Node *u) { if (p == nil) { r = u; // inserting into empty tree } else { int comp = compare(u->x, p->x); if (comp < 0) { p->left = u; } else if (comp > 0) { p->right = u; } else { return false; // u.x is already in the tree } u->parent = p; } n++; return true; } 109 6. Binary Trees 6.2. BinarySearchTree: An Unbalanced Binary Search Tree 7 7 3 11 3 11 1 5 9 13 1 5 9 13 4 6 8 12 14 4 6 8 12 14 8.5 Figure 6.7: Inserting the value 8.5 into a binary search tree. An example is shown in Figure 6.7. The most time-consuming part of this process is the initial search for x, which takes time proportional to the height of the newly added node u. In the worst case, this is equal to the height of the BinarySearchTree. 6.2.3 Deleting Deleting a value stored in a node, u, of a BinarySearchTree is a little more difficult. If u is a leaf, then we can just detach u from its parent. Even better: If u has only one child, then we can splice u from the tree by having u.parent adopt u’s child: BinarySearchTree void splice(Node *u) { Node *s, *p; if (u->left != nil) { s = u->left; } else { s = u->right; } if (u == r) { r = s; p = nil; } else { p = u->parent; if (p->left == u) { p->left = s; } else { p->right = s; } } if (s != nil) { s->parent = p; 110 6. Binary Trees 6.2. BinarySearchTree: An Unbalanced Binary Search Tree 7 3 11 1 5 9 13 4 6 8 12 14 Figure 6.8: Deleting a leaf (6) or a node with only one child (9) is easy. } n--; } Things get tricky, though, when u has two children. In this case, the simplest thing to do is to find a node, w, that has less than two children such that we can replace u.x with w.x. To maintain the binary search tree property, the value w.x should be close to the value of u.x. For example, picking w such that w.x is the smallest value greater than u.x will do. Finding the node w is easy; it is the smallest value in the subtree rooted at u.right. This node can be easily removed because it has no left child. (See Figure 6.9) BinarySearchTree void remove(Node *u) { if (u->left == nil || u->right == nil) { splice(u); delete u; } else { Node *w = u->right; while (w->left != nil) w = w->left; u->x = w->x; splice(w); delete w; } } 6.2.4 Summary The find(x), add(x), and remove(x) operations in a BinarySearchTree each involve fol- lowing a path from the root of the tree to some node in the tree. Without knowing more about the shape of the tree it is difficult to say much about the length of this path, except 111 6. Binary Trees 6.3. Discussion and Exercises 7 7 3 11 3 12 1 5 9 13 1 5 9 13 4 6 8 12 14 4 6 8 14 Figure 6.9: Deleting a value (11) from a node, u, with two children is done by replacing u’s value with the smallest value in the right subtree of u. that it is less than n, the number of nodes in the tree. The following (unimpressive) theorem summarizes the performance of the BinarySearchTree data structure: Theorem 6.1. A BinarySearchTree implements the SSet interface. A BinarySearchTree supports the operations add(x), remove(x), and find(x) in O(n) time per operation. Theorem 6.1 compares poorly with Theorem 4.1, which shows that the SkiplistSSet structure can implement the SSet interface with O(log n) expected time per operation. The problem with the BinarySearchTree structure is that it can become unbalanced. Instead of looking like the tree in Figure 6.5 it can look like a long chain of n nodes, all but the last having exactly one child. There are a number of ways of avoiding unbalanced binary search trees, all of which lead to data structures that have O(log n) time operations. In Chapter 7 we show how O(log n) expected time operations can be achieved with randomization. In Chapter 8 we show how O(log n) amortized time operations can be achieved with partial rebuilding oper- ations. In Chapter 9 we show how O(log n) worst-case time operations can be achieved by simulating a tree that is not binary: a tree in which nodes can have up to four children. 6.3 Discussion and Exercises Binary trees have been used to model relationships for literally thousands of years. One reason for this is that binary trees naturally model (pedigree) family trees. These are the family trees in which the root is a person, the left and right children are the person’s parents, and so on, recursively. In more recent centuries binary trees have also been used to model species-trees in biology, where the leaves of the tree represent extant species and the internal nodes of the tree represent speciation events in which two populations of a single species evolve into two separate species. 112 6. Binary Trees 6.3. Discussion and Exercises Binary search trees appear to have been discovered independently by several groups in the 1950s [41, Section 6.2.2]. Further references to specific kinds of binary search trees are provided in subsequent chapters. Exercise 6.1. Write a non-recursive variant of the size2() method, size(u), that computes the size of the subtree rooted at node u. Exercise 6.2. Write a non-recursive method, height2(u), that computes the height of node u in a binary search tree. Exercise 6.3. A pre-order traversal of a binary tree is a traversal that visits each node, u, before any of its children. An in-order traversal visits u after visiting all the nodes in u’s left subtree but before visiting any of the nodes in u’s right subtree. A post-order traversal visits u only after visiting all other nodes in u’s subtree. Write non-recursive functions nextPreOrder(u), nextInOrder(u), and nextPostOrder(u) that return the node that follows u in a pre-order, in-order, or post-order traversal, respec- tively. Exercise 6.4. Describe a sequence of n operations on an initially empty BinarySearchTree that results in a tree of height n − 1. Exercise 6.5. Design and implement a version of BinarySearchTree in which each node, u, maintains values u.size (the size of the subtree rooted at u), u.depth (the depth of u), and u.height (the height of the subtree rooted at u). These values should be maintained, even during the add(x) and remove(x) opera- tions, but this should not increase the cost of these operations by more than a constant factor. 113 6. Binary Trees 6.3. Discussion and Exercises 114 Chapter 7 Random Binary Search Trees In this chapter, we present a binary search tree structure that uses randomization to achieve O(log n) expected time for all operations. 7.1 Random Binary Search Trees Consider the two binary search trees shown in Figure 7.1. The one on the left is a list and the other is a perfectly balanced binary search tree. The one on the left has height n − 1 = 14 and the one on the right has height three. Imagine how these two trees could have been constructed. The one on the left occurs if we start with an empty BinarySearchTree and add the sequence h0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14i . No other sequence of additions will create this tree (as you can prove by induction on n). On the other hand, the tree on the right can be created by the sequence h7, 3, 11, 1, 5, 9, 13, 0, 2, 4, 6, 8, 10, 12, 14i . Other sequences work as well, including h7, 3, 1, 5, 0, 2, 4, 6, 11, 9, 13, 8, 10, 12, 14i , and h7, 3, 1, 11, 5, 0, 2, 4, 6, 9, 13, 8, 10, 12, 14i . In fact, there are 21, 964, 800 addition sequences that generate the tree on the right and only one that generates the tree on the left. The above example gives some anecdotal evidence that, if we choose a random permutation of 0, . . . , 14, and add it into a binary search tree then we are more likely to 115 7. Random Binary Search Trees 7.1. Random Binary Search Trees 0 1 2 7 3 3 11 .. . 1 5 9 13 14 0 2 4 6 8 10 12 14 Figure 7.1: Two binary search trees containing the integers 0, . . . , 14. get a very balanced tree (the right side of Figure 7.1) than we are to get a very unbalanced tree (the left side of Figure 7.1). We can formalize this notion by studying random binary search trees. A random binary search tree of size n is obtained in the following way: Take a random permutation x0 , . . . , xn−1 of 0, . . . , n − 1 and add its elements, one by one, into a BinarySearchTree. Note that the values 0, . . . , n − 1 could be replaced by any ordered set of n elements without changing any of the properties of the random binary search tree. The element x ∈ {0, . . . , n − 1} is simply standing in for the element of rank x in an ordered set of size n. Before we can present our main result about random binary search trees, we must take some time for a short digression to discuss a type of number that comes up frequently when studying randomized structures. For a non-negative integer, k, the k-th harmonic number, denoted Hk , is defined as Hk = 1 + 1/2 + 1/3 + · · · + 1/k . The harmonic number Hk has no simple closed form, but it is very closely related to the natural logarithm of k. In particular, ln k ≤ Hk ≤ ln k + 1 . Rk Readers who have studied calculus might notice that this is because the integral 1 (1/x) dx = ln k. Keeping in mind that an integral can be interpreted as the area between a curve and the Rk x-axis, the value of Hk can be lower-bounded by the integral 1 (1/x) dx and upper-bounded Rk by 1 + 1 (1/x) dx. (See Figure 7.2 for a graphical explanation.) 116 7. Random Binary Search Trees 7.1. Random Binary Search Trees 1 1 f (x) = 1/x 1/2 1/2 1/3 1/3 .. .. . . 1/k 1/k 0 1 2 3 ... k 1 2 3 ... k Pk Rk Figure 7.2: The kth harmonic number Hk = i=1 1/i is upper-bounded by 1 + 1 (1/x) dx Rk and lower-bounded by 1 (1/x) dx. Lemma 7.1. In a random binary search tree of size n, the following statements hold: 1. For any x ∈ {0, . . . , n − 1}, the expected length of the search path for x is Hx+1 + Hn−x − O(1).1 2. For any x ∈ (−1, n) \ {0, . . . , n − 1}, the expected length of the search path for x is Hdxe + Hn−dxe . We will prove Lemma 7.1 in the next section. For now, consider what the two parts of Lemma 7.1 tell us. The first part tells us that if we search for an element in a tree of size n, then the expected length of the search path is at most 2 ln n + O(1). The second part tells us the same thing about searching for a value not stored in the tree. When we compare the two parts of the lemma, we see that it is only slightly faster to search for something that is in a tree compared to something that is not in a tree. 7.1.1 Proof of Lemma 7.1 The key observation needed to prove Lemma 7.1 is the following: The search path for a value x in the open interval (−1, n) in a random binary search tree, T , contains the node with key i < x if and only if, in the random permutation used to create T , i appears before any of {i + 1, i + 2, . . . , bxc}. To see this, refer to Figure 7.3 and notice that, until some value in {i, i + 1, . . . , bxc} is added, the search paths for each value in the open interval (i − 1, bxc + 1) are identical. (Remember that for two search values to have different search paths, there must be some 1 The expressions x + 1 and n − x can be interpreted respectively as the number of elements in the tree less than or equal to x and the number of elements in the tree greater than or equal to x. 117 7. Random Binary Search Trees 7.1. Random Binary Search Trees j . . . , i, . . . , j − 1 j + 1, . . . , bxc, . . . Figure 7.3: The value i < x is on the search path for x if and only if i is the first element among {i, i + 1, . . . , bxc} added to the tree. element in the tree that compares differently with them.) Let j be the first element in {i, i + 1, . . . , bxc} to appear in the random permutation. Notice that j is now and will always be on the search path for x. If j 6= i then the node uj containing j is created before the node ui that contains i. Later, when i is added, it will be added to the subtree rooted at uj .left, since i < j. On the other hand, the search path for x will never visit this subtree because it will proceed to uj .right after visiting uj . Similarly, for i > x, i appears in the search path for x if and only if i appears before any of {dxe, dxe + 1, . . . , i − 1} in the random permutation used to create T . Notice that, if we start with a random permutation of {0, . . . , n}, then the subse- quences containing only {i, i + 1, . . . , bxc} and {dxe, dxe + 1, . . . , i − 1} are also random per- mutations of their respective elements. Each element, then, in the subsets {i, i + 1, . . . , bxc} and {dxe, dxe + 1, . . . , i − 1} is equally likely to appear before any other in its subset in the random permutation used to create T . So we have ( 1/(bxc − i + 1) if i < x Pr{i is on the search path for x} = . 1/(i − dxe + 1) if i > x With this observation, the proof of Lemma 7.1 involves some simple calculations with harmonic numbers: 118 7. Random Binary Search Trees 7.1. Random Binary Search Trees 1 1 1 1 1 1 1 Pr{Ii = 1} x+1 x ··· 3 2 2 3 ··· n−x i 0 1 ··· x−1 x x+1 ··· n−1 (a) 1 1 1 1 1 1 1 Pr{Ii = 1} bxc+1 bxc ··· 3 2 1 1 2 3 ··· n−bxc i 0 1 ··· bxc dxe ··· n−1 (b) Figure 7.4: The probabilities of an element being on the search path for x when (a) x is an integer and (b) when x is not an integer. Proof of Lemma 7.1. Let Ii be the indicator random variable that is equal to one when i appears on the search path for x and zero otherwise. Then the length of the search path is given by X Ii i∈{0,...,n−1}\{x} so, if x ∈ {0, . . . , n − 1}, the expected length of the search path is given by (see Figure 7.4.a) " x−1 n−1 # x−1 n−1 X X X X E Ii + Ii = E [Ii ] + E [Ii ] i=0 i=x+1 i=0 i=x+1 x−1 X n−1 X = 1/(bxc − i + 1) + 1/(i − dxe + 1) i=0 i=x+1 x−1 X n−1 X = 1/(x − i + 1) + 1/(i − x + 1) i=0 i=x+1 1 1 1 = + + ··· + 2 3 x+1 1 1 1 + + + ··· + 2 3 n−x = Hx+1 + Hn−x − 2 . The corresponding calculations for a search value x ∈ (−1, n) \ {0, . . . , n − 1} are almost identical (see Figure 7.4.b). 7.1.2 Summary The following theorem summarizes the performance of a random binary search tree: 119 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree 3, 1 1, 6 5, 11 0, 9 2, 99 4, 14 9, 17 7, 22 6, 42 8, 49 Figure 7.5: An example of a Treap containing the integers 0, . . . , 9. Each node, u, is illustrated with u.x, u.p. Theorem 7.1. A random binary search tree can be constructed in O(n log n) time. In a random binary search tree, the find(x) operation takes O(log n) expected time. 7.2 Treap: A Randomized Binary Search Tree The problem with random binary search trees is, of course, that they are not dynamic. They don’t support the add(x) or remove(x) operations needed to implement the SSet interface. In this section we describe a data structure called a Treap that uses Lemma 7.1 to implement the SSet interface. A node in a Treap is like a node in a BinarySearchTree in that it has a data value, x, but it also contains a unique numerical priority, p, that is assigned at random: Treap class TreapNode : public BSTNode<Node, T> { friend class Treap<Node,T>; int p; }; In addition to being a binary search tree, the nodes in a Treap also obey the heap property: At every node u, except the root, u.parent.p < u.p. That is, each node has a priority smaller than that of its two children. An example is shown in Figure 7.5. The heap and binary search tree conditions together ensure that, once the key (x) and priority (p) for each node are defined, the shape of the Treap is completely determined. The heap property tells us that the node with minimum priority has to be the root, r, of the Treap. The binary search tree property tells us that all nodes with keys smaller than 120 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree r.x are stored in the subtree rooted at r.left and all nodes with keys larger than r.x are stored in the subtree rooted at r.right. The important point about the priority values in a Treap is that they are unique and assigned at random. Because of this, there are two equivalent ways we can think about a Treap. As defined above, a Treap obeys the heap and binary search tree properties. Alternatively, we can think of a Treap as a BinarySearchTree whose nodes were added in increasing order of priority. For example, the Treap in Figure 7.5 can be obtained by adding the sequence of (x, p) values h(3, 1), (1, 6), (0, 9), (5, 11), (4, 14), (9, 17), (7, 22), (6, 42), (8, 49), (2, 99)i into a BinarySearchTree. Since the priorities are chosen randomly, this is equivalent to taking a random per- mutation of the keys — in this case the permutation is h3, 1, 0, 5, 9, 4, 7, 6, 8, 2i — and adding these to a BinarySearchTree. But this means that the shape of a treap is identical to that of a random binary search tree. In particular, if we replace each key x by its rank,2 then Lemma 7.1 applies. Restating Lemma 7.1 in terms of Treaps, we have: Lemma 7.2. In a Treap that stores a set S of n keys, the following statements hold: 1. For any x ∈ S, the expected length of the search path for x is Hr(x)+1 + Hn−r(x) − O(1). 2. For any x 6∈ S, the expected length of the search path for x is Hr(x) + Hn−r(x) . Here, r(x) denotes the rank of x in the set S ∪ {x}. Lemma 7.2 tells us that Treaps can implement the find(x) operation efficiently. However, the real benefit of a Treap is that it can support the add(x) and delete(x) operations. To do this, it needs to perform rotations in order to maintain the heap property. Refer to Figure 7.6. A rotation in a binary search tree is a local modification that takes a parent u of a node w and makes w the parent of u, while preserving the binary search tree property. Rotations come in two flavours: left or right depending on whether w is a right or left child of u, respectively. The code that implements this has to handle these two possibilities and be careful of a boundary case (when u is the root) so the actual code is a little longer than Figure 7.6 would lead a reader to believe: 2 The rank of an element x in a set S of elements is the number of elements in S that are less than x. 121 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree rotateRight(u) ⇒ u w ⇐ rotateLeft(w) w u C A A B B C Figure 7.6: Left and right rotations in a binary search tree. BinarySearchTree void rotateLeft(Node *u) { Node *w = u->right; w->parent = u->parent; if (w->parent != nil) { if (w->parent->left == u) { w->parent->left = w; } else { w->parent->right = w; } } u->right = w->left; if (u->right != nil) { u->right->parent = u; } u->parent = w; w->left = u; if (u == r) { r = w; r->parent = nil; } } void rotateRight(Node *u) { Node *w = u->left; w->parent = u->parent; if (w->parent != nil) { if (w->parent->left == u) { w->parent->left = w; } else { w->parent->right = w; } } 122 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree u->left = w->right; if (u->left != nil) { u->left->parent = u; } u->parent = w; w->right = u; if (u == r) { r = w; r->parent = nil; } } In terms of the Treap data structure, the most important property of a rotation is that the depth of w decreases by one while the depth of u increases by one. Using rotations, we can implement the add(x) operation as follows: We create a new node, u, and assign u.x = x and pick a random value for u.p. Next we add u using the usual add(x) algorithm for a BinarySearchTree, so that u is now a leaf of the Treap. At this point, our Treap satisfies the binary search tree property, but not necessarily the heap property. In particular, it may be the case that u.parent.p > u.p. If this is the case, then we perform a rotation at node w=u.parent so that u becomes the parent of w. If u continues to violate the heap property, we will have to repeat this, decreasing u’s depth by one every time, until u either becomes the root or u.parent.p < u.p. Treap bool add(T x) { Node *u = new Node; u->x = x; u->p = rand(); if (BinarySearchTree<Node,T>::add(u)) { bubbleUp(u); return true; } return false; } void bubbleUp(Node *u) { while (u->parent != nil && u->parent->p > u->p) { if (u->parent->right == u) { rotateLeft(u->parent); } else { rotateRight(u->parent); } } if (u->parent == nil) { r = u; } } 123 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree An example of an add(x) operation is shown in Figure 7.7. The running time of the add(x) operation is given by the time it takes to follow the search path for x plus the number of rotations performed to move the newly-added node, u, up to its correct location in the Treap. By Lemma 7.2, the expected length of the search path is at most 2 ln n + O(1). Furthermore, each rotation decreases the depth of u. This stops if u becomes the root, so the expected number of rotations cannot exceed the expected length of the search path. Therefore, the expected running time of the add(x) operation in a Treap is O(log n). (Exercise 7.3 asks you to show that the expected number of rotations performed during an insertion is actually only O(1).) The remove(x) operation in a Treap is the opposite of the add(x) operation. We search for the node, u, containing x and then perform rotations to move u downwards until it becomes a leaf and then we splice u from the Treap. Notice that, to move u downwards, we can perform either a left or right rotation at u, which will replace u with u.right or u.left, respectively. The choice is made by the first of the following that apply: 1. If u.left and u.right are both null, then u is a leaf and no rotation is performed. 2. If u.left (or u.right) is null, then perform a right (or left, respectively) rotation at u. 3. If u.left.p < u.right.p (or u.left.p > u.right.p), then perform a right rotation (or left rotation, respectively) at u. These three rules ensure that the Treap doesn’t become disconnected and that the heap property is maintained once u is removed. Treap bool remove(T x) { Node *u = findLast(x); if (u != nil && compare(u->x, x) == 0) { trickleDown(u); splice(u); delete u; return true; } return false; } void trickleDown(Node *u) { while (u->left != nil || u->right != nil) { if (u->left == nil) { rotateLeft(u); 124 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree 3, 1 1, 6 5, 11 0, 9 2, 99 4, 14 9, 14 1.5, 4 7, 22 6, 42 8, 49 3, 1 1, 6 5, 11 0, 9 1.5, 4 4, 14 9, 14 2, 99 7, 22 6, 42 8, 49 3, 1 1.5, 4 5, 11 1, 6 2, 99 4, 14 9, 14 0, 9 7, 22 6, 42 8, 49 Figure 7.7: Adding the value 1.5 into the Treap from Figure 7.5. 125 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree } else if (u->right == nil) { rotateRight(u); } else if (u->left->p < u->right->p) { rotateRight(u); } else { rotateLeft(u); } if (r == u) { r = u->parent; } } } An example of the remove(x) operation is shown in Figure 7.8. The trick to analyze the running time of the remove(x) operation is to notice that this operation is the reverse of the add(x) operation. In particular, if we were to reinsert x, using the same priority u.p, then the add(x) operation would do exactly the same number of rotations and would restore the Treap to exactly the same state it was in before the remove(x) operation took place. (Reading from bottom-to-top, Figure 7.8 illustrates the insertion of the value 9 into a Treap.) This means that the expected running time of the remove(x) on a Treap of size n is proportional to the expected running time of the add(x) operation on a Treap of size n−1. We conclude that the expected running time of remove(x) is O(log n). 7.2.1 Summary The following theorem summarizes the performance of the Treap data structure: Theorem 7.2. A Treap implements the SSet interface. A Treap supports the operations add(x), remove(x), and find(x) in O(log n) expected time per operation. It is worth comparing the Treap data structure to the SkiplistSSet data structure. Both implement the SSet operations in O(log n) expected time per operation. In both data structures, add(x) and remove(x) involve a search and then a constant number of pointer changes (see Exercise 7.3 below). Thus, for both these structures, the expected length of the search path is the critical value in assessing their performance. In a SkiplistSSet, the expected length of a search path is 2 log n + O(1) , 126 7. Random Binary Search Trees 7.2. Treap: A Randomized Binary Search Tree 3, 1 1, 6 5, 11 0, 9 2, 99 4, 14 9, 17 7, 22 6, 42 8, 49 3, 1 1, 6 5, 11 0, 9 2, 99 4, 14 7, 22 6, 42 9, 17 8, 49 3, 1 1, 6 5, 11 0, 9 2, 99 4, 14 7, 22 6, 42 8, 49 9, 17 3, 1 1, 6 5, 11 0, 9 2, 99 4, 14 7, 22 6, 42 8, 49 Figure 7.8: Removing the value 9 from the Treap in Figure 7.5. 127 7. Random Binary Search Trees 7.3. Discussion and Exercises In a Treap, the expected length of a search path is 2 ln n + O(1) ≈ 1.386 log n + O(1) . Thus, the search paths in a Treap are considerably shorter and this translates into noticeably faster operations on Treaps than Skiplists. Exercise 4.2 in Chapter 4 shows how the expected length of the search path in a Skiplist can be reduced to e ln n + O(1) ≈ 1.884 log n + O(1) by using biased coin tosses. Even with this optimization, the expected length of search paths in a SkiplistSSet is noticeably longer than in a Treap. 7.3 Discussion and Exercises Random binary search trees have been studied extensively. Devroye [14] gives a proof of Lemma 7.1 and related results. There are much stronger results in the literature as well. The most impressive of which is due to Reed [53], who shows that the expected height of a random binary search tree is α ln n − β ln ln n + O(1) where α ≈ 4.31107 is the unique solution on [2, ∞) of the equation α ln((2e/α)) = 1 and 3 β= 2 ln(α/2) . Furthermore, the variance of the height is constant. The name Treap was coined by Aragon and Seidel [56] who discussed Treaps and some of their variants. However, their basic structure was studied much earlier by Vuillemin [61] who called them Cartesian trees. One space-optimization of the Treap data structure that is sometimes performed is the elimination of the explicit storage of the priority p in each node. Instead, the prior- ity of a node, u, is computed by hashing u’s address in memory. Although a number of hash functions will probably work well for this in practice, for the important parts of the proof of Lemma 7.1 to remain valid, the hash function should be randomized and have the min-wise independent property: For any distinct values x1 , . . . , xk , each of the hash values h(x1 ), . . . , h(xk ) should be distinct with high probability and, for each i ∈ {1, . . . , k}, Pr{h(xi ) = min{h(x1 ), . . . , h(xk )}} ≤ c/k for some constant c. One such class of hash functions that is easy to implement and fairly fast is tabulation hashing [51]. 128 7. Random Binary Search Trees 7.3. Discussion and Exercises Exercise 7.1. Prove the assertion that there are 21, 964, 800 sequences that generate the tree on the right hand side of Figure 7.1. (Hint: Give a recursive formula for the number of sequences that generate a complete binary tree of height h and evaluate this formula for h = 3.) Exercise 7.2. Design and implement the permute(a) method that takes as input an array, a, containing n distinct values and randomly permutes a. The method should run in O(n) time and you should prove that each of the n! possible permutations of a is equally probable. Exercise 7.3. Use both parts of Lemma 7.2 to prove that the expected number of rotations performed by an add(x) operation (and hence also a remove(x) operation) is O(1). Exercise 7.4. Design and implement a version of a Treap that includes a get(i) operation that returns the key with rank i in the Treap. (Hint: Have each node, u, keep track of the size of the subtree rooted at u.) Exercise 7.5. Design and implement a version of a Treap that supports the split(x) oper- ation. This operation removes all values from the Treap that are greater than x and returns a second Treap that contains all the removed values. For example, the code t2 = t.split(x) removes from t all values greater than x and returns a new Treap t2 containing all these values. The split(x) operation should run in O(log n) expected time. 129 7. Random Binary Search Trees 7.3. Discussion and Exercises 130 Chapter 8 Scapegoat Trees In this chapter, we study a binary search tree data structure, the ScapegoatTree, that keeps itself balanced by partial rebuilding operations. During a partial rebuilding operation, an entire subtree is deconstructed and rebuilt into a perfectly balanced subtree. There are many ways of rebuilding a subtree rooted at node u into a perfectly balanced tree. One of the simplest is to traverse u’s subtree, gathering all its nodes into an array a and then to recursively build a balanced subtree using a. If we let m = a.length/2, then the element a[m] becomes the root of the new subtree, a[0], . . . , a[m − 1] get stored recursively in the left subtree and a[m + 1], . . . , a[a.length − 1] get stored recursively in the right subtree. ScapegoatTree void rebuild(Node *u) { int ns = BinaryTree<Node>::size(u); Node *p = u->parent; Node **a = new Node*[ns]; packIntoArray(u, a, 0); if (p == nil) { r = buildBalanced(a, 0, ns); r->parent = nil; } else if (p->right == u) { p->right = buildBalanced(a, 0, ns); p->right->parent = p; } else { p->left = buildBalanced(a, 0, ns); p->left->parent = p; } delete[] a; } int packIntoArray(Node *u, Node **a, int i) { if (u == nil) { return i; 131 8. Scapegoat Trees 8.1. ScapegoatTree: A Binary Search Tree with Partial Rebuilding } i = packIntoArray(u->left, a, i); a[i++] = u; return packIntoArray(u->right, a, i); } A call to rebuild(u) takes O(size(u)) time. The subtree built by rebuild(u) has minimum height; there is no tree of smaller height that has size(u) nodes. 8.1 ScapegoatTree: A Binary Search Tree with Partial Rebuilding A ScapegoatTree is a BinarySearchTree that, in addition to keeping track of the number, n, of nodes in the tree also keeps a counter, q, that maintains an upper-bound on the number of nodes. ScapegoatTree int q; At all times, n and q obey the following inequalities: q/2 ≤ n ≤ q . In addition, a ScapegoatTree has logarithmic height; at all times, the height of the scape- goat tree does not exceed: log3/2 q ≤ log3/2 2n < log3/2 n + 2 . (8.1) Even with this constraint, a ScapegoatTree can look surprisingly unbalanced. The tree in Figure 8.1 has q = n = 10 and height 5 < log3/2 10 ≈ 5.679. Implementing the find(x) operation in a ScapegoatTree is done using the stan- dard algorithm for searching in a BinarySearchTree (see Section 6.2). This takes time proportional to the height of the tree which, by (8.1) is O(log n). To implement the add(x) operation, we first increment n and q and then use the usual algorithm for adding x to a binary search tree; we search for x and then add a new leaf u with u.x = x. At this point, we may get lucky and the depth of u might not exceed log3/2 q. If so, then we leave well enough alone and don’t do anything else. Unfortunately, it will sometimes happen that depth(u) > log3/2 q. In this case we need to do something to reduce the height. This isn’t a big job; there is only one node, namely u, whose depth exceeds log3/2 q. To fix u, we walk from u back up to the root 132 8. Scapegoat Trees 8.1. ScapegoatTree: A Binary Search Tree with Partial Rebuilding 7 6 8 5 9 2 1 4 0 3 Figure 8.1: A ScapegoatTree with 10 nodes and height 5. looking for a scapegoat, w. The scapegoat, w, is a very unbalanced node. It has the property that size(w.child) 2 > , (8.2) size(w) 3 where w.child is the child of w on the path from the root to u. We’ll very shortly prove that a scapegoat exists. For now, we can take it for granted. Once we’ve found the scapegoat w, we completely destroy the subtree rooted at w and rebuild it into a perfectly balanced binary search tree. We know, from (8.2), that, even before the addition of u, w’s subtree was not a complete binary tree. Therefore, when we rebuild w, the height decreases by at least 1 so that height of the ScapegoatTree is once again at most log3/2 q. ScapegoatTree bool add(T x) { // first do basic insertion keeping track of depth Node *u = new Node; u->x = x; u->left = u->right = u->parent = nil; int d = addWithDepth(u); if (d > log32(q)) { // depth exceeded, find scapegoat Node *w = u->parent; int a = BinaryTree<Node>::size(w); int b = BinaryTree<Node>::size(w->parent); while (3*a <= 2*b) { w = w->parent; a = BinaryTree<Node>::size(w); 133 8. Scapegoat Trees 8.1. ScapegoatTree: A Binary Search Tree with Partial Rebuilding 7 7 6 8 6 8 6 2 5 7 > 3 9 3 9 3 2 6 1 4 2 1 4 3 0 2 3.5 5 1 0 3 2 3.5 Figure 8.2: Inserting 3.5 into a ScapegoatTree increases its depth to 6, which violates (8.1) since 6 > log3/2 11 ≈ 5.914. A scapegoat is found at the node containing 5. b = BinaryTree<Node>::size(w->parent); } rebuild(w->parent); } return d >= 0; } If we ignore the cost of finding the scapegoat w and rebuilding the subtree rooted at w, then the running time of add(x) is dominated by the initial search, which takes O(log q) = O(log n) time. We will account for the cost of finding the scapegoat and rebuilding using amortized analysis in the next section. The implementation of remove(x) in a ScapegoatTree is very simple. We search for x and remove it using the usual algorithm for removing a node from a BinarySearchTree. (Note that this can never increase the height of the tree.) Next, we decrement n but leave q unchanged. Finally, we check if q > 2n and, if so, we rebuild the entire tree into a perfectly balanced binary search tree and set q = n. ScapegoatTree bool remove(T x) { if (BinarySearchTree<Node,T>::remove(x)) { if (2*n < q) { rebuild(r); q = n; } 134 8. Scapegoat Trees 8.1. ScapegoatTree: A Binary Search Tree with Partial Rebuilding return true; } return false; } Again, if we ignore the cost of rebuilding, the running time of the remove(x) operation is proportional to the height of the tree, and is therefore O(log n). 8.1.1 Analysis of Correctness and Running-Time In this section we analyze the correctness and amortized running time of operations on a ScapegoatTree. We first prove the correctness by showing that, when the add(x) operation results in a node that violates Condition (8.1), then we can always find a scapegoat: Lemma 8.1. Let u be a node of depth h > log3/2 q in a ScapegoatTree. Then there exists a node w on the path from u to the root such that size(w) > 2/3 . size(parent(w)) Proof. Suppose, for the sake of contradiction, that this is not the case, and size(w) ≤ 2/3 . size(parent(w)) for all nodes w on the path from u to the root. Denote the path from the root to u as r = u0 , . . . , uh = u. Then, we have size(u0 ) = n, size(u1 ) ≤ 23 n, size(u2 ) ≤ 49 n and, more generally, i 2 size(ui ) ≤ n . 3 But this gives a contradiction, since size(u) ≥ 1, hence h log3/2 q log3/2 n 2 2 2 1 1 ≤ size(u) ≤ n< n≤ n= n=1 . 3 3 3 n Next, we analyze the parts of the running time that we have not yet accounted for. There are two parts: The cost of calls to size(u) when search for scapegoat nodes, and the cost of calls to rebuild(w) when we find a scapegoat w. The cost of calls to size(u) can be related to the cost of calls to rebuild(w), as follows: Lemma 8.2. During a call to add(x) in a ScapegoatTree, the cost of finding the scapegoat w and rebuilding the subtree rooted at w is O(size(w)). 135 8. Scapegoat Trees 8.1. ScapegoatTree: A Binary Search Tree with Partial Rebuilding Proof. The cost of rebuilding the scapegoat node w, once we find it, is O(size(w)). When searching for the scapegoat node, we call size(u) on a sequence of nodes u0 , . . . , uk until we find the scapegoat uk = w. However, since uk is the first node in this sequence that is a scapegoat, we know that 2 size(ui ) < size(ui+1 ) 3 for all i ∈ {0, . . . , k − 2}. Therefore, the cost of all calls to size(u) is k ! k−1 ! X X O size(uk−i ) = O size(uk ) + size(uk−i−1 ) i=0 i=0 k−1 i ! X 2 = O size(uk ) + size(uk ) 3 i=0 k−1 i !! X 2 = O size(uk ) 1 + 3 i=0 = O(size(uk )) = O(size(w)) , where the last line follows from the fact that the sum is a geometrically decreasing series. All that remains is to prove an upper-bound on the cost of calls to rebuild(u): Lemma 8.3. Starting with an empty ScapegoatTree any sequence of m add(x) and remove(x) operations causes at most O(m log m) time to be used by rebuild(u) operations. Proof. To prove this, we will use a credit scheme. Each node stores a number of credits. Each credit can pay for some constant, c, units of time spent rebuilding. The scheme gives out a total of O(m log m) credits and every call to rebuild(u) is paid for with credits stored at u. During an insertion or deletion, we give one credit to each node on the path to the inserted node, or deleted node, u. In this way we hand out at most log3/2 q ≤ log3/2 m credits per operation. During a deletion we also store an additional 1 credit “on the side.” Thus, in total we give out at most O(m log m) credits. All that remains is to show that these credits are sufficient to pay for all calls to rebuild(u). If we call rebuild(u) during an insertion, it is because u is a scapegoat. Suppose, without loss of generality, that size(u.left) 2 > . size(u) 3 136 8. Scapegoat Trees 8.2. Discussion and Exercises Using the fact that size(u) = 1 + size(u.left) + size(u.right) we deduce that 1 size(u.left) > size(u.right) 2 and therefore 1 1 size(u.left) − size(u.right) > size(u.left) > size(u) . 2 3 Now, the last time a subtree containing u was rebuilt (or when u was inserted, if a subtree containing u was never rebuilt), we had size(u.left) − size(u.right) ≤ 1 . Therefore, the number of add(x) or remove(x) operations that have affected u.left or u.right since then is at least 1 size(u) − 1 . 3 and there are therefore at least this many credits stored at u that are available to pay for the O(size(u)) time it takes to call rebuild(u). If we call rebuild(u) during a deletion, it is because q > 2n. In this case, we have q − n > n credits stored “on the side” and we use these to pay for the O(n) time it takes to rebuild the root. This completes the proof. 8.1.2 Summary The following theorem summarizes the performance of the ScapegoatTree data structure: Theorem 8.1. A ScapegoatTree implements the SSet interface. Ignoring the cost of rebuild(u) operations, a ScapegoatTree supports the operations add(x), remove(x), and find(x) in O(log n) time per operation. Furthermore, beginning with an empty ScapegoatTree, any sequence of m add(x) and remove(x) operations results in a total of O(m log m) time spent during all calls to rebuild(u). 8.2 Discussion and Exercises The term scapegoat tree is due to Galperin and Rivest [28], who define and analyze these trees. However, the same structure was discovered earlier by Andersson [3, 5], who called them general balanced trees since they can have any shape as long as their height is small. 137 8. Scapegoat Trees 8.2. Discussion and Exercises Experimenting with the ScapegoatTree implementation will reveal that it is of- ten considerably slower than the other SSet implementations in this book. This may be somewhat surprising, since height bound of log3/2 q ≈ 1.709 log n + O(1) is better than the expected length of a search path in a Skiplist and not too far from that of a Treap. The implementation could be optimized by storing the sizes of subtrees explicitly at each node (or at least reusing already computed subtree sizes). Even with these optimizations, there will always be sequences of add(x) and delete(x) operation for which a ScapegoatTree takes longer than other SSet implementations. This gap in performance is due to the fact that, unlike the other SSet implemen- tations discussed in this book, a ScapegoatTree can spend a lot of time restructuring itself. Exercise 8.1 asks you to prove that there are sequences of n operations in which a ScapegoatTree will spend on the order of n log n time in calls to rebuild(u). This is in con- trast to other SSet implementations discussed in this book that only make O(n) structural changes during a sequence of n operations. This is, unfortunately, a necessary consequence of the fact that a ScapegoatTree does all its restructuring by calls to rebuild(u) [15]. Despite their lack of performance, there are applications in which a ScapegoatTree could be the right choice. This would occur any time there is additional data associated with nodes that cannot be updated in constant time when a rotation is performed, but that can be updated during a rebuild(u) operation. In such cases, the ScapegoatTree and related structures based on partial rebuilding may work. An example of such an application is outlined in Exercise 8.5. Exercise 8.1. Show that, if we start with an empty ScapegoatTree and call add(x) for x = 1, 2, 3, . . . , n, then the total time spent during calls to rebuild(u) is at least cn log n for some constant c > 0. Exercise 8.2. The ScapegoatTree, as described in this chapter guarantees that the length of the search path does not exceed log3/2 q. Design, analyze, and implement a modified version of ScapegoatTree where the length of the search path does not exceed logb q, where b is a parameter with 1 < b < 2. What does your analysis and/or experiments say about the amortized cost of find(x), add(x) and remove(x) as a function of b? Exercise 8.3. Analyze and implement a WeightBalancedTree. This is a tree in which each node u, except the root, maintains the balance invariant that size(u) ≤ (2/3)size(u.parent). 138 8. Scapegoat Trees 8.2. Discussion and Exercises The add(x) and remove(x) operations are identical to the standard BinarySearchTree op- erations, except that any time the balance invariant is violated at a node u, the subtree rooted at u.parent is rebuilt. Your analysis should show that operations on a WeightBalancedTree run in O(log n) amortized time. Exercise 8.4. Analyze and implement a CountdownTree. In a CountdownTree each node u keeps a timer u.t. The add(x) and remove(x) operations are exactly the same as in a standard BinarySearchTree except that, whenever one of these operations affects u’s subtree, u.t is decremented. When u.t = 0 the entire subtree rooted at u is rebuilt into a perfectly balanced binary search tree. When a node u is involved in a rebuilding operation (either because u is rebuilt or one of u’s ancestors is rebuilt) u.t is reset to size(u)/3. Your analysis should show that operations on a countdown tree run in O(log n) amortized time. (Hint: First show that each node u satisfies some version of a balance invariant.) Exercise 8.5. Design and implement a Sequence data structure that maintains a sequence (list) of elements. It supports these operations: • addAfter(e): Add a new element after the element e in the sequence. Return the newly added element. (If e is null, the new element is added at the beginning of the sequence.) • remove(e): Remove e from the sequence. • testBefore(e1, e2): return true if and only if e1 comes before e2 in the sequence. The first two operations should run in O(log n) amortized time. The third operation should run in constant-time. The Sequence data structure can be implemented by storing the elements in some- thing like a ScapegoatTree, in the same order that they occur in the sequence. To imple- ment testBefore(e1, e2) in constant time, each element e is labelled with an integer that encodes the path from the root to e. In this way, testBefore(e1, e2) can be implemented just by comparing the labels of e1 and e2. 139 8. Scapegoat Trees 8.2. Discussion and Exercises 140 Chapter 9 Red-Black Trees In this chapter, we present red-black trees, a version of binary search trees that have loga- rithmic depth. Red-black trees are one of the most widely-used data structures in practice. They appear as the primary search structure in many library implementations, including the Java Collections Framework and several implementations of the C++ Standard Tem- plate Library. They are also used within the Linux operating system kernel. There are several reasons for the popularity of red-black trees: 1. A red-black tree storing n values has height at most 2 log n. 2. The add(x) and remove(x) operations on a red-black tree run in O(log n) worst-case time. 3. The amortized number of rotations done during an add(x) or remove(x) operation is constant. The first two of these properties already put red-black trees ahead of skiplists, treaps, and scapegoat trees. Skiplists and treaps rely on randomization and their O(log n) running times are only expected. Scapegoat trees have a guaranteed bound on their height, but add(x) and remove(x) only run in O(log n) amortized time. The third property is just icing on the cake. It tells us that that the time needed to add or remove an element x is dwarfed by the time it takes to find x.1 However, the nice properties of red-black trees come with a price: implementation complexity. Maintaining a bound of 2 log n on the height is not easy. It requires a careful analysis of a number of cases and it requires that the implementation does exactly the right 1 Note that skiplists and treaps also have this property in the expected sense. See Exercise 4.1 and Exercise 7.3. 141 9. Red-Black Trees 9.1. 2-4 Trees Figure 9.1: A 2-4 tree of height 3. thing in each case. One misplaced rotation or change of color produces a bug that can be very difficult to understand and track down. Rather than jumping directly into the implementation of red-black trees, we will first provide some background on a related data structure: 2-4 trees. This will give some insight into how red-black trees were discovered and why efficiently maintaining red-black trees is even possible. 9.1 2-4 Trees A 2-4 tree is a rooted tree with the following properties: Property 9.1 (height). All leaves have the same depth. Property 9.2 (degree). Every internal node has 2, 3, or 4 children. An example of a 2-4 tree is shown in Figure 9.1. The properties of 2-4 trees imply that their height is logarithmic in the number of leaves: Lemma 9.1. A 2-4 tree with n leaves has height at most log n. Proof. The lower-bound of 2 on the degree of an internal node implies that, if the height of a 2-4 tree is h, then it has at least 2h leaves. In other words, n ≥ 2h . Taking logarithms on both sides of this equation gives h ≤ log n. 9.1.1 Adding a Leaf Adding a leaf to a 2-4 tree is easy (see Figure 9.2). If we want to add a leaf u as the child of some node w on the second-last level, we simply make u a child of w. This certainly 142 9. Red-Black Trees 9.1. 2-4 Trees w w u w w0 u Figure 9.2: Adding a leaf to a 2-4 Tree. This process stops after one split because w.parent has degree less than 4 before the addition. maintains the height property, but could violate the degree property; if w had 4 children prior to adding u, then w now has 5 children. In this case, we split w into two nodes, w and w’, having 2 and 3 children, respectively. But now w’ has no parent, so we recursively make w’ a child of w’s parent. Again, this may cause w’s parent to have too many children in which case we split it. This process goes on until we reach a node that has fewer than 4 children, or until we split the root, r, into two nodes r and r0 . In the latter case, we make a new root that has r and r0 as children. This simultaneously increases the depth of all leaves and so maintains the height property. Since the height of the 2-4 tree is never more than log n, the process of adding a leaf finishes after at most log n steps. 143 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree 9.1.2 Removing a Leaf Removing a leaf from a 2-4 tree is a little more tricky (see Figure 9.3). To remove a leaf u from its parent w, we just remove it. If w had only two children prior to the removal of u, then w is left with only one child and violates the degree property. To correct this, we look at w’s sibling, w0 . The node w0 is sure to exist since w’s parent has at least 2 children. If w0 has 3 or 4 children, then we take one of these children from w0 and give it to w. Now w has 2 children and w0 has 2 or 3 children and we are done. On the other hand, if w0 has only two children, then we merge w and w0 into a single node, w, that has 3 children. Next we recursively remove w0 from the parent of w0 . This process ends when we reach a node, u, where u or its sibling has more than 2 children; or we reach the root. In the latter case, if the root is left with only 1 child, then we delete the root and make its child the new root. Again, this simultaneously decreases the height of every leaf and therefore maintains the height property. Again, since the height of the tree is never more than log n, the process of removing a leaf finishes after at most log n steps. 9.2 RedBlackTree: A Simulated 2-4 Tree A red-black tree is a binary search tree in which each node, u, has a color which is either red or black. Red is represented by the value 0 and black by the value 1. RedBlackTree class RedBlackNode : public BSTNode<Node, T> { friend class RedBlackTree<Node, T>; char color; }; int red = 0; int black = 1; Before and after any operation on a red-black tree, the following two properties are satisfied. Each property is defined both in terms of the colors red and black, and in terms of the numeric values 0 and 1. Property 9.3 (black-height). There are the same number of black nodes on every root to leaf path. (The sum of the colors on any root to leaf path is the same.) Property 9.4 (no-red-edge). No two red nodes are adjacent. (For any node u, except the root, u.color + u.parent.color ≥ 1.) Notice that we can always color the root, r, of a red-black tree black without vi- olating either of these two properties, so we will assume that the root is black, and the 144 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree u Figure 9.3: Removing a leaf from a 2-4 Tree. This process goes all the way to the root because all of u’s ancestors and their siblings have degree 2. 145 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree Figure 9.4: An example of a red-black tree with black-height 3. External (nil) nodes are drawn as squares. algorithms for updating a red-black tree will maintain this. Another trick that simplifies red-black trees is to treat the external nodes (represented by nil) as black nodes. This way, every real node, u, of a red-black tree has exactly two children, each with a well-defined color. An example of a red-black tree is shown in Figure 9.4. 9.2.1 Red-Black Trees and 2-4 Trees At first it might seem surprising that a red-black tree can be efficiently updated to maintain the black-height and no-red-edge properties, and it seems unusual to even consider these as useful properties. However, red-black trees were designed to be an efficient simulation of 2-4 trees as binary trees. Refer to Figure 9.5. Consider any red-black tree, T , having n nodes and perform the following transformation: Remove each red node u and connect u’s two children directly to the (black) parent of u. After this transformation we are left with a tree T 0 having only black nodes. Every internal node in T 0 has 2, 3, or 4 children: A black node that started out with two black children will still have two black children after this transformation. A black node that started out with one red and one black child will have three children after this transformation. A black node that started out with two red children will have 4 children after this transformation. Furthermore, the black-height property now guarantees that every root-to-leaf path in T 0 has the same length. In other words, T 0 is a 2-4 tree! The 2-4 tree T 0 has n + 1 leaves that correspond to the n + 1 external nodes of the red-black tree. Therefore, this tree has height log(n + 1). Now, every root to leaf path in the 2-4 tree corresponds to a path from the root of the red-black tree T to an external node. The first and last node in this path are black and at most one out of every two internal 146 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree Figure 9.5: Every red-black tree has a corresponding 2-4 tree. nodes is red, so this path has at most log(n + 1) black nodes and at most log(n + 1) − 1 red nodes. Therefore, the longest path from the root to any internal node in T is at most 2 log(n + 1) − 2 ≤ 2 log n , for any n ≥ 1. This proves the most important property of red-black trees: Lemma 9.2. The height of red-black tree with n nodes is at most 2 log n. Now that we have seen the relationship between 2-4 trees and red-black trees, it is not hard to believe that we can efficiently maintain a red-black tree while adding and removing elements. We have already seen that adding an element in a BinarySearchTree can be done by adding a new leaf. Therefore, to implement add(x) in a red-black tree we need a method of simulating splitting a degree 5 node in a 2-4 tree. A degree 5 node is represented by a black node that has two red children one of which also has a red child. We can “split” this node by coloring it red and coloring its two children black. An example of this is shown in Figure 9.6. Similarly, implementing remove(x) requires a method of merging two nodes and borrowing a child from a sibling. Merging two nodes is the inverse of a split (shown in Figure 9.6), and involves coloring two (black) siblings red and coloring their (red) parent 147 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree w w u w w0 u Figure 9.6: Simulating a 2-4 tree split operation during an addition in a red-black tree. (This simulates the 2-4 tree addition shown in Figure 9.2.) 148 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree black. Borrowing from a sibling is the most complicated of the procedures and involves both rotations and recoloring of nodes. Of course, during all of this we must still maintain the no-red-edge property and the black-height property. While it is no longer surprising that this can be done, there are a large number of cases that have to be considered if we try to do a direct simulation of a 2-4 tree by a red-black tree. At some point, it just becomes simpler to forget about the underlying 2-4 tree and work directly towards maintaining the red-black tree properties. 9.2.2 Left-Leaning Red-Black Trees There is no single definition of a red-black tree. Rather, there are a family of structures that manage to maintain the black-height and no-red-edge properties during add(x) and remove(x) operations. Different structures go about it in different ways. Here, we implement a data structure that we call a RedBlackTree. This structure implements a particular variant of red-black trees that satisfies an additional property: Property 9.5 (left-leaning). At any node u, if u.left is black, then u.right is black. Note that the red-black tree shown in Figure 9.4 does not satisfy the left-leaning property; it is violated by the parent of the red node in the rightmost path. The reason for maintaining the left-leaning property is that it reduces the number of cases encountered when updating the tree during add(x) and remove(x) operations. In terms of 2-4 trees, it implies that every 2-4 tree has a unique representation: A node of degree 2 becomes a black node with 2 black children. A node of degree 3 becomes a black node whose left child is red and whose right child is black. A node of degree 4 becomes a black node with two red children. Before we describe the implementation of add(x) and remove(x) in detail, we first present some simple subroutines used by these methods that are illustrated in Figure 9.7. The first two subroutines are for manipulating colors while preserving the black-height prop- erty. The pushBlack(u) method takes as input a black node u that has two red children and colors u red and its two children black. The pullBlack(u) method reverses this operation: RedBlackTree void pushBlack(Node *u) { u->color--; u->left->color++; u->right->color++; } void pullBlack(Node *u) { u->color++; 149 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree u u u u pushBlack(u) pullBlack(u) flipLeft(u) flipRight(u) ⇓ ⇓ ⇓ ⇓ u u u u Figure 9.7: Flips, pulls and pushes u->left->color--; u->right->color--; } The flipLeft(u) method swaps the colors of u and u.right and then performs a left rotation at u. This reverses the colors of these two nodes as well as their parent-child relationship: RedBlackTree void flipLeft(Node *u) { swapColors(u, u->right); rotateLeft(u); } The flipLeft(u) operation is especially useful in restoring the left-leaning property at a node u that violates it (because u.left is black and u.right is red). In this special case, we can be assured this operation preserves both the black-height and no-red-edge properties. The flipRight(u) operation is symmetric to flipLeft(u) with the roles of left and right reversed. RedBlackTree void flipRight(Node *u) { swapColors(u, u->left); rotateRight(u); } 9.2.3 Addition To implement add(x) in a RedBlackTree, we perform a standard BinarySearchTree inser- tion, which adds a new leaf, u, with u.x = x and set u.color = red. Note that this does 150 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree not change the black height of any node, so it does not violate the black-height property. It may, however, violate the left-leaning property (if u is the right child of its parent) and it may violate the no-red-edge property (if u’s parent is red). To restore these properties, we call the method addFixup(u). RedBlackTree bool add(T x) { Node *u = new Node(); u->left = u->right = u->parent = nil; u->x = x; u->color = red; bool added = BinarySearchTree<Node,T>::add(u); if (added) addFixup(u); return added; } The addFixup(u) method, illustrated in Figure 9.8, takes as input a node u whose color is red and which may be violating the no-red-edge property and/or the left-leaning property. The following discussion is probably impossible to follow without referring to Figure 9.8 or recreating it on a piece of paper. Indeed, the reader may wish to study this figure before continuing. If u is the root of the tree, then we can color u black and this restores both properties. If u’s sibling is also red, then u’s parent must be black, so both the left-leaning and no-red- edge properties already hold. Otherwise, we first determine if u’s parent, w, violates the left-leaning property and, if so, perform a flipLeft(w) operation and set u = w. This leaves us in a well-defined state: u is the left child of its parent, w, so w now satisfies the left-leaning property. All that remains is to ensure the no-red-edge property at u. We only have to worry about the case where w is red, since otherwise u already satisfies the no-red-edge property. Since we are not done yet, u is red and w is red. The no-red-edge property (which is only violated by u and not by w) implies that u’s grandparent g exists and is black. If g’s right child is red, then the left-leaning property ensures that both g’s children are red, and a call to pushBlack(g) makes g red and w black. This restores the no-red-edge property at u, but may cause it to be violated at g, so the whole process starts over with u = g. If g’s right child is black, then a call to flipRight(g) makes w the (black) parent of g and gives w two red children, u and g. This ensures that u satisfies the no-red-edge property and g satisfies the left-leaning property. In this case we can stop. 151 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree u u.parent.left.color w w w u u u return flipLeft(w) ; u = w w u w.color w w u u g.right.color return g g g w w w u u u flipRight(g) pushBlack(g) pushBlack(g) w new u new u u w w u u return Figure 9.8: A single round in the process of fixing Property 2 after an insertion. 152 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree RedBlackTree void addFixup(Node *u) { while (u->color == red) { if (u == r) { // u is the root - done u->color = black; return; } Node *w = u->parent; if (w->left->color == black) { // ensure left-leaning flipLeft(w); u = w; w = u->parent; } if (w->color == black) return; // no red-red edge = done Node *g = w->parent; // grandparent of u if (g->right->color == black) { flipRight(g); return; } else { pushBlack(g); u = g; } } } The insertFixup(u) method takes constant time per iteration and each iteration either finishes or moves u closer to the root. This implies that the insertFixup(u) method finishes after O(log n) iterations in O(log n) time. 9.2.4 Removal The remove(x) operation in a RedBlackTree tree is the most complicated operation to implement, and this is true of all known implementations. Like remove(x) in a Binary- SearchTree, this operation boils down to finding a node w with only one child, u, and splicing w out of the tree by having w.parent adopt u. The problem with this is that, if w is black, then the black-height property will now be violated at w.parent. We get around this problem, temporarily, by adding w.color to u.color. Of course, this introduces two other problems: (1) u and w both started out black, then u.color + w.color = 2 (double black), which is an invalid color. If w was red, then it is replaced by a black node u, which may violate the left-leaning property at u.parent. Both of these problems are resolved with a call to the removeFixup(u) method. 153 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree RedBlackTree bool remove(T x) { Node *u = findLast(x); if (u == nil || compare(u->x, x) != 0) return false; Node *w = u->right; if (w == nil) { w = u; u = w->left; } else { while (w->left != nil) w = w->left; u->x = w->x; u = w->right; } splice(w); u->color += w->color; u->parent = w->parent; delete w; removeFixup(u); return true; } The removeFixup(u) method takes as input a node u whose color is black (1) or double-black (2). If u is double-black, then removeFixup(u) performs a series of rotations and recoloring operations that move the double-black node up the tree until it can be gotten rid of. During this process, the node u changes until, at the end of this process, u refers to the root of the subtree that has been changed. The root of this subtree may have changed color. In particular, it may have gone from red to black, so the removeFixup(u) method finishes by checking if u’s parent violates the left-leaning property and, if so, fixes it. RedBlackTree void removeFixup(Node *u) { while (u->color > black) { if (u == r) { u->color = black; } else if (u->parent->left->color == red) { u = removeFixupCase1(u); } else if (u == u->parent->left) { u = removeFixupCase2(u); } else { u = removeFixupCase3(u); } } 154 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree if (u != r) { // restore left-leaning property, if necessary Node *w = u->parent; if (w->right->color == red && w->left->color == black) { flipLeft(w); } } } The removeFixup(u) method is illustrated in Figure 9.9. Again, the following text will be very difficult, if not impossible, to follow without referring constantly to Figure 9.9. Each iteration of the loop in removeFixup(u) processes the double-black node u based on one of four cases. Case 0: u is the root. This is the easiest case to treat. We recolor u to be black and this does not violate any of the red-black tree properties. Case 1: u’s sibling, v, is red. In this case, u’s sibling is the left child of its parent, w (by the left-leaning property). We perform a right-flip at w and then proceed to the next iteration. Note that this causes w’s parent to violate the left-leaning property and it causes the depth of u to increase. However, it also implies that the next iteration will be in Case 3 with w colored red. When examining Case 3, below, we will see that this means the process will stop during the next iteration. RedBlackTree Node* removeFixupCase1(Node *u) { flipRight(u->parent); return u; } Case 2: u’s sibling, v, is black and u is the left child of its parent, w. In this case, we call pullBlack(w), making u black, v red, and darkening the color of w to black or double-black. At this point, w does not satisfy the left-leaning property, so we call flipLeft(w) to fix this. At this point, w is red and v is the root of the subtree we started with. We need to check if w causes no-red-edge property to be violated. We do this by inspecting w’s right child, q. If q is black, then w satisfies the no-red-edge property and we can continue to the next iteration with u=v. Otherwise (q is red), both the no-red-edge property and the left-leaning property are violated at q and w, respectively. A call to rotateLeft(w) restores the left-leaning property, but the no-red-edge property is still violated. At this point, q is the left child of v and w is the left child of q, q and w are both red and v is black or double-black. A flipRight(v) 155 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree removeFixupCase2(u) removeFixupCase3(u) removeFixupCase1(u) w w w u v v u u pullBlack(w) pullBlack(w) flipRight(w) w w u v v u flipLeft(w) flipRight(w) w new u v v w w u q q.color q.color v v (new u) v v w w w u q q q u q u v.left.color rotateLeft(w) rotateRight(w) v v v w w q q q u q u w w flipLeft(v) pushBlack(v) u u flipRight(v) flipLeft(v) w (new u) v u w q q q u w v v w u u pushBlack(q) pushBlack(q) q q w v v w u u v.right.color q q w v w v u u flipLeft(v) q w u v Figure 9.9: A single round in the process of eliminating a double-black node after a removal. 156 9. Red-Black Trees 9.2. RedBlackTree: A Simulated 2-4 Tree makes q the parent of both v and w. Following this up by a pushBlack(q) makes both v and w black and sets the color of q back to the original color of w. At this point, there is no more double-black node and the no-red-edge and black- height properties are reestablished. The only possible problem that remains is that the right child of v may be red, in which case the left-leaning property is violated. We check this and perform a flipLeft(v) to correct it if necessary. RedBlackTree Node* removeFixupCase2(Node *u) { Node *w = u->parent; Node *v = w->right; pullBlack(w); // w->left flipLeft(w); // w is now red Node *q = w->right; if (q->color == red) { // q-w is red-red rotateLeft(w); flipRight(v); pushBlack(q); if (v->right->color == red) flipLeft(v); return q; } else { return v; } } Case 3: u’s sibling is black and u is the right child of its parent, w. This case is symmetric to Case 2 and is handled mostly the same way. The only differences come from the fact that the left-leaning property is asymmetric, so it requires different handling. As before, we begin with a call to pullBlack(w), which makes v red and u black. A call to flipRight(w) promotes v to the root of the subtree. At this point w is red, and the code branches two ways depending on the color of w’s left child, q. If q is red, then the code finishes up exactly the same way that Case 2 finishes up, but is even simpler since there is no danger of v not satisfying the left-leaning property. The more complicated case occurs when q is black. In this case, we examine the color if v’s left child. If it is red, then v has two red children and its extra black can be pushed down with a call to pushBlack(v). At this point, v now has w’s original color and we are done. If v’s left child is black then v violates the left-leaning property and we restore this with a call to flipLeft(v). The next iteration of removeFixup(u) then continues with 157 9. Red-Black Trees 9.3. Summary u = v. RedBlackTree Node* removeFixupCase3(Node *u) { Node *w = u->parent; Node *v = w->left; pullBlack(w); flipRight(w); // w is now red Node *q = w->left; if (q->color == red) { // q-w is red-red rotateRight(w); flipLeft(v); pushBlack(q); return q; } else { if (v->left->color == red) { pushBlack(v); // both v’s children are red return v; } else { // ensure left-leaning flipLeft(v); return w; } } } Each iteration of removeFixup(u) takes constant time. Cases 2 and 3 either finish or move u closer to the root of the tree. Case 0 (where u is the root) always terminates and Case 1 leads immediately to Case 3, which also terminates. Since the height of the tree is at most 2 log n, we conclude that there are at most O(log n) iterations of removeFixup(u) so removeFixup(u) runs in O(log n) time. 9.3 Summary The following theorem summarizes the performance of the RedBlackTree data structure: Theorem 9.1. A RedBlackTree implements the SSet interface. A RedBlackTree supports the operations add(x), remove(x), and find(x) in O(log n) worst-case time per operation. Not included in the above theorem is the extra bonus Theorem 9.2. Beginning with an empty RedBlackTree, any sequence of m add(x) and remove(x) operations results in a total of O(m) time spent during all calls addFixup(u) and removeFixup(u). 158 9. Red-Black Trees 9.4. Discussion and Exercises We will only sketch a proof of Theorem 9.2. By comparing addFixup(u) and removeFixup(u) with the algorithms for adding or removing a leaf in a 2-4 tree, we can convince ourselves that this property is something that is inherited from a 2-4 tree. In particular, if we can show that the total time spent splitting, merging, and borrowing in a 2-4 tree is O(m), then this implies Theorem 9.2. The proof of this for 2-4 trees uses the potential method of amortized analysis.2 Define the potential of an internal node u in a 2-4 tree as 1 if u has 2 children Φ(u) = 0 if u has 3 children 3 if u has 4 children and the potential of a 2-4 tree as the sum of the potentials of its nodes. When a split occurs, it is because a node of degree 4 becomes two nodes, one of degree 2 and one of degree 3. This means that the overall potential drops by 3 − 1 − 0 = 2. When a merge occurs, two nodes that used to have degree 2 are replaced by one node of degree 3. The result is a drop in potential of 2 − 0 = 2. Therefore, for every split or merge, the potential decreases by 2. Next notice that, if we ignore splitting and merging of nodes, there are only a constant number of nodes whose number of children is changed by the addition or removal of a leaf. When adding a node, one node has its number of children increase by 1, increasing the potential by at most 3. During the removal of a leaf, one node has its number of children decrease by 1, increasing the potential by at most 1, and two nodes may be involved in a borrowing operation, increasing their total potential by at most 1. To summarize, each merge and split causes the potential to drop by at least 2. Ignoring merging and splitting, each addition or removal causes the potential to rise by at most 3, and the potential is always non-negative. Therefore, the number of splits and merges caused by m additions or removals on an initially empty tree is at most 3m/2. Theorem 9.2 is a consequence of this analysis and the correspondence between 2-4 trees and red-black trees. 9.4 Discussion and Exercises Red-black trees were first introduced by Guibas and Sedgewick [32]. Despite their high implementation complexity they are found in some of the most commonly used libraries and applications. Most algorithms and data structures discuss some variant of red-black 2 See the proofs of Lemma 2.2 and Lemma 3.1 for other applications of the potential method. 159 9. Red-Black Trees 9.4. Discussion and Exercises trees. Andersson [4] describes a left-leaning version of balanced trees that are similar to red-black trees but have the additional constraint that any node has at most one red child. This implies that these trees simulate 2-3 trees rather than 2-4 trees. They are significantly simpler, though, than the RedBlackTree structure presented in this chapter. Sedgewick [55] describes at least two versions of left-leaning red-black trees. These use recursion along with a simulation of top-down splitting and merging in 2-4 trees. The combination of these two techniques makes for particularly short and elegant code. A related, and older, data structure is the AVL tree [2]. AVL trees are height- balanced : At each node u, the height of the subtree rooted at u.left and the subtree rooted at u.right differ by at most one. It follows immediately that, if F (h) is the minimum number of leaves in a tree of height h, then F (h) obeys the Fibonacci recurrence F (h) = F (h − 1) + F (h − 2) √ with base cases F (0) = 1 and F (1) = 1. This means F (h) is approximately ϕh / 5, where √ √ ϕ = (1 + 5)/2 ≈ 1.61803399 is the golden ratio. (More precisely, |ϕh / 5 − F (h)| ≤ 1/2.) Arguing as in the proof of Lemma 9.1, this implies h ≤ logϕ n ≈ 1.440420088 log n , so AVL trees have smaller height than red-black trees. The height-balanced property can be maintained during add(x) and remove(x) operations by walking back up the path to the root and performing a rebalancing operation at each node u where the height of u’s left and right subtrees differ by 2. See Figure 9.10. Andersson’s variant of red-black trees, Sedgewick’s variant of red-black trees, and AVL trees are all simpler to implement than the RedBlackTree structure defined here. Unfortunately, none of them can guarantee that the amortized time spent rebalancing is O(1) per update. In particular, there is no analogue of Theorem 9.2 for those structures. Exercise 9.1. Why does the method remove(x) in the RedBlackTree implementation per- form the assignment u.parent = w.parent? Shouldn’t this already be done by the call to splice(w)? Exercise 9.2. Suppose a 2-4 tree, T , has n` leaves and ni internal nodes. 1. What is the minimum value of ni , as a function of n` ? 2. What is the maximum value of ni , as a function of n` ? 160 9. Red-Black Trees 9.4. Discussion and Exercises h h+2 h h+2 h+1 Figure 9.10: Rebalancing in an AVL tree. At most 2 rotations are required to convert a node whose subtrees have height h and h + 2 into a node whose subtrees each have height at most h + 1. 161 9. Red-Black Trees 9.4. Discussion and Exercises 3. If T 0 is a red-black tree that represents T , then how many red nodes does T 0 have? Exercise 9.3. Prove that, during an add(x) operation, an AVL tree must perform at most one rebalancing operation (that involves at most 2 rotations; see Figure 9.10). Give an example of an AVL tree and a remove(x) operation on that tree that requires on the order of log n rebalancing operations. Exercise 9.4. Implement an AVLTree class that implements AVL trees as described above. Compare its performance to that of the RedBlackTree implementation. Which implemen- tation has a faster find(x) operation? Exercise 9.5. Design and implement a series of experiments that compare the relative per- formance of find(x), add(x), and remove(x) for SkiplistSSet, ScapegoatTree, Treap, and RedBlackTree. Be sure to include multiple test scenarios, including cases where the data is random, already sorted, is removed in random order, is removed in sorted order, and so on. 162 Chapter 10 Heaps In this chapter, we discuss two implementations of the extremely useful priority Queue data structure. The first is an implementation based on arrays. It is very fast and is the basis of one of the fastest known sorting algorithms, namely heapsort (see Section 11.1.3). The second implementation is based on binary trees and is more flexible. In particular, it supports a meld(h) operation that allows the priority queue to absorb the elements of a second priority queue h. 10.1 BinaryHeap: An Implicit Binary Tree Our first implementation of a (priority) Queue is based on a technique that is over 400 years old. Eytzinger’s method allows us to represent a complete binary tree as an array. This is done by laying out the nodes of the tree in breadth-first order (see Section 6.1.2) in the array. In this way, the root is stored at position 0, the root’s left child is stored at position 1, the root’s right child at position 2, the left child of the left child of the root is stored at position 3, and so on. See Figure 10.1. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 10.1: Eytzinger’s method represents a complete binary tree as an array. 163 10. Heaps 10.1. BinaryHeap: An Implicit Binary Tree If we do this for a large enough tree, some patterns emerge. The left child of the node at index i is at index left(i) = 2i + 1 and the right child of the node at index i is at index right(i) = 2i + 2. The parent of the node at index i is at index parent(i) = (i − 1)/2. BinaryHeap int left(int i) { return 2*i + 1; } int right(int i) { return 2*i + 2; } int parent(int i) { return (i-1)/2; } A BinaryHeap uses this technique to implicitly represent a complete binary tree in which the elements are heap-ordered : The value stored at any index i is not smaller than the value stored at index parent(i), with the exception of the root value, i = 0. It follows that the smallest value in the priority Queue is therefore stored at position 0 (the root). In a BinaryHeap, the n elements are stored in an array a: BinaryHeap array<T> a; int n; Implementing the add(x) operation is fairly straightforward. As with all array-based structures, we first check if a is full (because a.length = n) and, if so, we grow a. Next, we place x at location a[n] and increment n. At this point, all that remains is to ensure that we maintain the heap property. We do this by repeatedly swapping x with its parent until x is no longer smaller than its parent. See Figure 10.2. BinaryHeap bool add(T x) { if (n + 1 > a.length) resize(); a[n++] = x; bubbleUp(n-1); return true; } void bubbleUp(int i) { int p = parent(i); while (i > 0 && compare(a[i], a[p]) < 0) { a.swap(i,p); i = p; p = parent(i); 164 10. Heaps 10.1. BinaryHeap: An Implicit Binary Tree } } Implementing the remove() operation, which removes the smallest value from the heap, is a little trickier. We know where the smallest value is (at the root), but we need to replace it after we remove it and ensure that we maintain the heap property. The easiest way to do this is to replace the root with the value a[n] and decrement n. Unfortunately, the new element now at the root is probably not the smallest element, so it needs to be moved downwards. We do this by repeatedly comparing this element to its two children. If it is the smallest of the three then we are done. Otherwise, we swap this element with the smallest of its two children and continue. BinaryHeap T remove() { T x = a[0]; a[0] = a[--n]; trickleDown(0); if (3*n < a.length) resize(); return x; } void trickleDown(int i) { do { int j = -1; int r = right(i); if (r < n && compare(a[r], a[i]) < 0) { int l = left(i); if (compare(a[l], a[r]) < 0) { j = l; } else { j = r; } } else { int l = left(i); if (l < n && compare(a[l], a[i]) < 0) { j = l; } } if (j >= 0) a.swap(i, j); i = j; } while (i >= 0); } As with other array-based structures, we will ignore the time spent in calls to 165 10. Heaps 10.1. BinaryHeap: An Implicit Binary Tree 4 9 8 17 26 50 16 19 69 32 93 55 4 9 8 17 26 50 16 19 69 32 93 55 4 9 8 17 26 50 16 19 69 32 93 55 6 4 9 8 17 26 50 16 19 69 32 93 55 6 4 9 8 17 26 6 16 19 69 32 93 55 50 4 9 8 17 26 6 16 19 69 32 93 55 50 4 9 6 17 26 8 16 19 69 32 93 55 50 4 9 6 17 26 8 16 19 69 32 93 55 50 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 10.2: Adding the value 6 to a BinaryHeap. 166 10. Heaps 10.1. BinaryHeap: An Implicit Binary Tree 4 9 6 17 26 8 16 19 69 32 93 55 50 4 9 6 17 26 8 16 19 69 32 93 55 50 50 9 6 17 26 8 16 19 69 32 93 55 50 9 6 17 26 8 16 19 69 32 93 55 6 9 50 17 26 8 16 19 69 32 93 55 6 9 50 17 26 8 16 19 69 32 93 55 6 9 8 17 26 50 16 19 69 32 93 55 6 9 8 17 26 50 16 19 69 32 93 55 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Figure 10.3: Removing the minimum value, 4, from a BinaryHeap. 167 10. Heaps 10.2. MeldableHeap: A Randomized Meldable Heap resize(), since these can be accounted for with the amortization argument from Lemma 2.1. The running times of both add(x) and remove() then depend on the height of the (implicit) binary tree. However, this is a complete binary tree; every level except the last has the maximum possible number of nodes. Therefore, if the height of this tree is h, then it has at least 2h nodes. Stated another way n ≥ 2h . Taking logarithms on both sides of this equation gives h ≤ log n . Therefore, both the add(x) and remove() operation run in O(log n) time. 10.1.1 Summary The following theorem summarizes the performance of a BinaryHeap: Theorem 10.1. A BinaryHeap implements the (priority) Queue interface. Ignoring the cost of calls to resize(), a BinaryHeap supports the operations add(x) and remove() in O(log n) time per operation. Furthermore, beginning with an empty BinaryHeap, any sequence of m add(x) and remove() operations results in a total of O(m) time spent during all calls to resize(). 10.2 MeldableHeap: A Randomized Meldable Heap In this section, we describe the MeldableHeap, a priority Queue implementation in which the underlying structure is also a heap-ordered binary tree. However, unlike a BinaryHeap in which the underlying binary tree is completely defined by the number of elements, there are no restrictions on the shape of the binary tree that underlies a MeldableHeap; anything goes. The add(x) and remove() operations in a MeldableHeap are implemented in terms of the merge(h1, h2) operation. This operation takes two heap nodes h1 and h2 and merges them, returning a heap node that is the root of a heap that contains all elements in the subtree rooted at h1 and all elements in the subtree rooted at h2. The nice thing about a merge(h1, h2) operation is that it can be defined recursively. If either of h1 or h2 is nil, then we are merging with an empty set, so we return h2 or h1, respectively. Otherwise, assume h1.x ≤ h2.x since, if h1.x > h2.x, then we can reverse the roles of h1 and h2. Then we know that the root of the merged heap will contain h1.x 168 10. Heaps 10.2. MeldableHeap: A Randomized Meldable Heap and we can recursively merge h2 with h1.left or h1.right, as we wish. This is where randomization comes in, and we toss a coin to decide whether to merge h2 with h1.left or h1.right: MeldableHeap Node* merge(Node *h1, Node *h2) { if (h1 == nil) return h2; if (h2 == nil) return h1; if (compare(h1->x, h2->x) > 0) return merge(h2, h1); // now we know h1->x <= h2->x if (rand() % 2) { h1->left = merge(h1->left, h2); if (h1->left != nil) h1->left->parent = h1; } else { h1->right = merge(h1->right, h2); if (h1->right != nil) h1->right->parent = h1; } return h1; } In the next section, we show that merge(h1, h2) runs in O(log n) expected time, where n is the total number of elements in h1 and h2. With access to a merge(h1, h2) operation, the add(x) operation is easy. We create a new node u containing x and then merge u with the root of our heap: MeldableHeap bool add(T x) { Node *u = new Node(); u->left = u->right = u->parent = nil; u->x = x; r = merge(u, r); r->parent = nil; n++; return true; } This takes O(log(n + 1)) = O(log n) expected time. The remove() operation is similarly easy. The node we want to remove is the root, so we just merge its two children and make the result the root: MeldableHeap T remove() { T x = r->x; Node *tmp = r; r = merge(r->left, r->right); 169 10. Heaps 10.2. MeldableHeap: A Randomized Meldable Heap delete tmp; if (r != nil) r->parent = nil; n--; return x; } Again, this takes O(log n) expected time. Additionally, a MeldableHeap can implement many other operations in O(log n) expected time, including: • remove(u): remove the node u (and its key u.x) from the heap. • absorb(h): add all the elements of the MeldableHeap h to this heap, emptying h in the process. Each of these operations can be implemented using a constant number of merge(h1, h2) operations that each take O(log n) time. 10.2.1 Analysis of merge(h1, h2) The analysis of merge(h1, h2) is based on the analysis of a random walk in a binary tree. A random walk in a binary tree is a walk that starts at the root of the tree. At each step in the walk, a coin is tossed and the walk proceeds to the left or right child of the current node depending on the result of this coin toss. The walk ends when it falls off the tree (the current node becomes nil). The following lemma is somewhat remarkable because it does not depend at all on the shape of the binary tree: Lemma 10.1. The expected length of a random walk in a binary tree with n nodes is at most log(n + 1). Proof. The proof is by induction on n. In the base case, n = 0 and the walk has length 0 = log(n + 1). Suppose now that the result is true for all non-negative integers n0 < n. Let n1 denote the size of the root’s left subtree, so that n2 = n − n1 − 1 is the size of the root’s right subtree. Starting at the root, the walk takes one step and then continues in a subtree of size n1 or continues in a subtree of size n2 . By our inductive hypothesis, the expected length of the walk is then 1 1 E[W ] = 1 + log(n1 + 1) + log(n2 + 1) , 2 2 170 10. Heaps 10.2. MeldableHeap: A Randomized Meldable Heap since each of n1 and n2 are less than n. Since log is a concave function, E[W ] is maximized when n1 = n2 = (n − 1)/2. Therefore, the expected number of steps taken by the random walk is 1 1 E[W ] = 1 + log(n1 + 1) + log(n2 + 1) 2 2 ≤ 1 + log((n − 1)/2 + 1) = 1 + log((n + 1)/2) = log(n + 1) . We make a quick digression to note that, for readers who know a little about infor- mation theory, the proof of Lemma 10.1 can be stated in terms of entropy. Information Theoretic Proof of Lemma 10.1. Let di denote the depth of the ith external node and recall that a binary tree with n nodes has n + 1 external nodes. The probability of the random walk reaching the ith external node is exactly pi = 1/2di , so the expected length of the random walk is given by n X n X X n H= pi di = pi log 2di = pi log(1/pi ) i=0 i=0 i=0 The right hand side of this equation is easily recognizable as the entropy of a probability distribution over n + 1 elements. A basic fact about the entropy of a distribution over n + 1 elements is that it does not exceed log(n + 1), which proves the lemma. With this result on random walks, we can now easily prove that the running time of the merge(h1, h2) operation is O(log n). Lemma 10.2. If h1 and h2 are the roots of two heaps containing n1 and n2 nodes, respec- tively, then the expected running time of merge(h1, h2) is at most O(log n), where n = n1 +n2 . Proof. Each step of the merge algorithm takes one step of a random walk, either in the heap rooted at h1 or the heap rooted at h2 The algorithm terminates when either of these two random walks fall out of its corresponding tree (when h1 = null or h2 = null). Therefore, the expected number of steps performed by the merge algorithm is at most log(n1 + 1) + log(n2 + 1) ≤ 2 log n . 171 10. Heaps 10.3. Discussion and Exercises 10.2.2 Summary The following theorem summarizes the performance of a MeldableHeap: Theorem 10.2. A MeldableHeap implements the (priority) Queue interface. A MeldableHeap supports the operations add(x) and remove() in O(log n) expected time per operation. 10.3 Discussion and Exercises The implicit representation of a complete binary tree as an array, or list, seems to have been first proposed by Eytzinger [22], as a representation for pedigree family trees. The BinaryHeap data structure described here was first introduced by Williams [63]. The randomized MeldableHeap data structure described here appears to have first been proposed by Gambin and Malinowski [29]. Other meldable heap implementations exist, including leftist heaps [12, 41, Section 5.3.2], binomial heaps [60], Fibonacci heaps [27], pairing heaps [26], and skew heaps [59], although none of these are as simple as the MeldableHeap structure. Some of the above structures also support a decreaseKey(u, y) operation in which the value stored at node u is decreased to y. (It is a pre-condition that y ≤ u.x.) This operation can be implemented in O(log n) time in most of the above structures by removing node u and adding y. However, some of these structures can implement decreaseKey(u, y) more efficiently. In particular, decreaseKey(u, y) takes O(1) amortized time in Fibonacci heaps and O(log log n) amortized time in a special version of pairing heaps [20]. This more efficient decreaseKey(u, y) operation has applications in speeding up several graph algorithms including Dijkstra’s shortest path algorithm [27]. 172 Chapter 11 Sorting Algorithms This chapter discusses algorithms for sorting a set of n items. This might seem like a strange topic for a book on data structures, but there are several good reasons for including it here. The most obvious reason is that two of these sorting algorithms (quicksort and heap-sort) are intimately related to two of the data structures we have already studied (random binary search trees and heaps, respectively). The first part of this chapter discusses algorithms that sort using only comparisons and presents three algorithms that run in O(n log n) time. As it turns out, all three al- gorithms are asymptotically optimal; no algorithm that uses only comparisons can avoid doing roughly n log n comparisons in the worst-case and even the average-case. The second part of this chapter shows that, if we allow other operations besides comparisons, then all bets are off. Indeed, by using array-indexing, it is possible to sort a set of n integers in the range {0, . . . , nc − 1} in O(cn) time. 11.1 Comparison-Based Sorting In this section, we present three sorting algorithms: merge-sort, quicksort, and heap-sort. All these algorithms take an input array a and sort the elements of a into non-decreasing order in O(n log n) (expected) time. These algorithms are all comparison-based. These algorithms don’t care what type of data is being sorted, the only operation they do on the data is comparisons using the compare(a, b) method. Recall, from Section 1.1.4, that compare(a, b) returns a negative value if a < b, a positive value if a > b, and zero if a = b. 11.1.1 Merge-Sort The merge-sort algorithm is a classic example of recursive divide and conquer: If the length of a is at most 1, then a is already sorted, so we do nothing. Otherwise, we split a into two halves, a0 = a[0], . . . , a[n/2 − 1] and a1 = a[n/2], . . . , a[n − 1]. We recursively sort a0 and 173 11. Sorting Algorithms 11.1. Comparison-Based Sorting a 13 8 5 2 4 0 6 9 7 3 12 1 10 11 a0 13 8 5 2 4 0 6 9 7 3 12 1 10 11 a1 mergeSort(a0, c) mergeSort(a1, c) a0 0 2 4 5 6 8 13 1 3 7 9 10 11 12 a1 merge(a0, a1, a) a 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Figure 11.1: The execution of mergeSort(a, c) a1, and then we merge (the now sorted) a0 and a1 to get our fully sorted array a: Algorithms void mergeSort(array<T> &a) { if (a.length <= 1) return; array<T> a0(0); array<T>::copyOfRange(a0, a, 0, a.length/2); array<T> a1(0); array<T>::copyOfRange(a1, a, a.length/2, a.length); mergeSort(a0); mergeSort(a1); merge(a0, a1, a); } An example is shown in Figure 11.1. Compared to sorting, merging the two sorted arrays a0 and a1 fairly easy. We add elements to a one at a time. If a0 or a1 is empty we add the next element from the other (non-empty) array. Otherwise, we take the minimum of the next element in a0 and the next element in a1 and add it to a: Algorithms void merge(array<T> &a0, array<T> &a1, array<T> &a) { int i0 = 0, i1 = 0; for (int i = 0; i < a.length; i++) { if (i0 == a0.length) a[i] = a1[i1++]; else if (i1 == a1.length) a[i] = a0[i0++]; else if (compare(a0[i0], a1[i1]) < 0) 174 11. Sorting Algorithms 11.1. Comparison-Based Sorting a[i] = a0[i0++]; else a[i] = a1[i1++]; } } Notice that the merge(a0, a1, a, c) algorithm performs at most n − 1 comparisons before running out of elements in one of a0 or a1. To understand the running-time of merge-sort, it is easiest to think of it in terms of its recursion tree. Suppose for now that n is a power of 2, so that n = 2log n , and log n is an integer. Refer to Figure 11.2. Merge-sort turns the problem of sorting n elements into 2 problems, each of sorting n/2 elements. These two subproblem are then turned into 2 problems each, for a total of 4 subproblems, each of size n/4. These 4 subproblems become 8 subproblems of size n/8, and so on. At the bottom of this process, n/2 subproblems, each of size 2, are converted into n problems, each of size 1. For each subproblem of size n/2i , the time spent merging and copying data is O(n/2i ). Since there are 2i subproblems of size n/2i , the total time spent working on problems of size 2i , not counting recursive calls, is 2i × O(n/2i ) = O(n) . Therefore, the total amount of time taken by merge-sort is Xn log O(n) = O(n log n) . i=0 The proof of the following theorem is based on the same analysis as above, but has to be a little more careful to deal with the cases where n is not a power of 2. Theorem 11.1. The mergeSort(a, c) algorithm runs in O(n log n) time and performs at most n log n comparisons. Proof. The proof is by induction on n. The base case, in which n = 1, is trivial. Merging two sorted lists of total length n requires at most n − 1 comparisons. Let C(n) denote the maximum number of comparisons performed by mergeSort(a, c) on an ar- ray a of length n. If n is even, then we apply the inductive hypothesis to the two subproblems 175 11. Sorting Algorithms 11.1. Comparison-Based Sorting n =n n n 2 2 =n n n n n 4 + 4 + 4 + 4 =n n n n n n n n n 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 =n .. .. .. .. .. .. .. .. . . . . . . . . 2 + 2 + 2 + ··· + 2 + 2 + 2 =n 1 + 1 + 1 + 1 + 1 + 1 + ··· + 1 + 1 + 1 + 1 + 1 + 1 = n Figure 11.2: The merge-sort recursion tree. and obtain C(n) ≤ n − 1 + 2C(n/2) ≤ n − 1 + 2((n/2) log(n/2)) = n − 1 + n log(n/2) = n − 1 + n log n − n < n log n . The case where n is odd is slightly more complicated. For this case, we use two inequalities, that are easy to verify: log(x + 1) ≤ log(x) + 1 , (11.1) for all x ≥ 1 and log(x + 1/2) + log(x − 1/2) ≤ 2 log(x) , (11.2) for all x ≥ 1/2. Inequality (11.1) comes from the fact that log(x) + 1 = log(2x) while (11.2) follows from the fact that log is a concave function. With these tools in hand we have, for 176 11. Sorting Algorithms 11.1. Comparison-Based Sorting odd n, C(n) ≤ n − 1 + C(dn/2e) + C(bn/2c) ≤ n − 1 + dn/2e logdn/2e + bn/2c logbn/2c = n − 1 + (n/2 + 1/2) log(n/2 + 1/2) + (n/2 − 1/2) log(n/2 − 1/2) ≤ n − 1 + n log(n/2) + (1/2)(log(n/2 + 1/2) − log(n/2 − 1/2)) ≤ n − 1 + n log(n/2) + 1/2 < n + n log(n/2) = n + n(log n − 1) = n log n . 11.1.2 Quicksort The quicksort algorithm is another classic divide and conquer algorithm. Unlike merge-sort, which does merging after solving the two subproblems, quicksort does all its work upfront. The algorithm is simple to describe: Pick a random pivot element, x, from a; par- tition a into the set of elements less than x, the set of elements equal to x, and the set of elements greater than x; and, finally, recursively sort the first and third sets in this partition. An example is shown in Figure 11.3. Algorithms void quickSort(array<T> &a) { quickSort(a, 0, a.length); } void quickSort(array<T> &a, int i, int n) { if (n <= 1) return; T x = a[i + rand()%n]; int p = i-1, j = i, q = i+n; // a[i..p]<x, a[p+1..q-1]??x, a[q..i+n-1]>x while (j < q) { int comp = compare(a[j], x); if (comp < 0) { // move to beginning of array a.swap(j++, ++p); } else if (comp > 0) { a.swap(j, --q); // move to end of array } else { j++; // keep in the middle } } // a[i..p]<x, a[p+1..q-1]=x, a[q..i+n-1]>x quickSort(a, i, p-i+1); 177 11. Sorting Algorithms 11.1. Comparison-Based Sorting x 13 8 5 2 4 0 6 9 7 3 12 1 10 11 1 8 5 2 4 0 6 7 3 9 12 10 11 13 quickSort(a, 0, 9) quickSort(a, 10, 4) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Figure 11.3: An example execution of quickSort(a, 0, 14) quickSort(a, q, n-(q-i)); } All of this is done in-place, so that instead of making copies of subarrays being sorted, the quickSort(a, i, n, c) method only sorts the subarray a[i], . . . , a[i + n − 1]. Initially, this method is called as quickSort(a, 0, a.length, c). At the heart of the quicksort algorithm is the in-place partitioning that, without any extra space, swaps elements in a and computes indices p and q so that < x if 0 ≤ i ≤ p a[i] = x if p < i < q > x if q ≤ i ≤ n − 1 This partitioning, which is done by the while loop in the code, works by iteratively increas- ing p and decreasing q while maintaining the first and last of these conditions. At each step, the element at position j is either moved to the front, left where it is, or moved to the back. In the first two cases, j is incremented, while in the last case, j is not incremented since the new element at position j has not been processed yet. Quicksort is very closely related to the random binary search trees studied in Sec- tion 7.1. In fact, if the input to quicksort consists of n distinct elements, then the quicksort recursion tree is a random binary search tree. To see this, recall that when constructing a random binary search tree the first thing we do is pick a random element x and make it the root of the tree. After this, every element will eventually be compared to x, with smaller elements going into the left subtree and larger elements going into the right subtree. 178 11. Sorting Algorithms 11.1. Comparison-Based Sorting In quicksort, we select a random element x and immediately compare everything to x, putting the smaller elements at the beginning of the array and larger elements at the end of the array. Quicksort then recursively sorts the beginning of the array and the end of the array, while the random binary search tree recursively inserts smaller elements in the left subtree of the root and larger elements in the right subtree of the root. The above correspondence between random binary search trees and quicksort means that we can translate Lemma 7.1 to a statement about quicksort: Lemma 11.1. When quicksort is called to sort an array containing the integers 0, . . . , n−1, the expected number of times element i is compared to a pivot element is at most Hi+1 + Hn−i . A little summing of harmonic numbers gives us the following theorem about the running time of quicksort: Theorem 11.2. When quicksort is called to sort an array containing n distinct elements, the expected number of comparisons performed is 2n ln n + O(n). Proof. Let T be the number of comparisons performed by quicksort when sorting n distinct elements. Using Lemma 11.1, we have: n−1 X E[T ] = (Hi+1 + Hn−i ) i=0 Xn =2 Hi i=1 Xn ≤2 Hn i=1 ≤ 2n ln n + 2n = 2n ln n + O(n) Theorem 11.3 describes the case where the elements being sorted are all distinct. When the input array, a, contains duplicate elements, the expected running time of quicksort is no worse, and can be even better; any time a duplicate element x is chosen as a pivot, all occurrences of x get grouped together and don’t take part in either of the two subproblems. Theorem 11.3. The quickSort(a, c) method runs in O(n log n) expected time and the expected number of comparisons it performs is at most 2n ln n + O(n). 179 11. Sorting Algorithms 11.1. Comparison-Based Sorting 5 9 6 10 13 8 7 11 12 5 9 6 10 13 8 7 11 12 4 3 2 1 0 0 1 2 3 4 5 6 7 8 11 12 13 14 15 Figure 11.4: A snapshot of the execution of heapSort(a, c). The shaded part of the array is already sorted. The unshaded part is a BinaryHeap. During the next iteration, element 5 will be placed into array location 8. 11.1.3 Heap-sort The heap-sort algorithm is another in-place sorting algorithm. Heap-sort uses the binary heaps discussed in Section 10.1. Recall that the BinaryHeap data structure represents a heap using a single array. The heap-sort algorithm converts the input array a into a heap and then repeatedly extracts the minimum value. More specifically, a heap stores n elements at array locations a[0], . . . , a[n − 1] with the smallest value stored at the root, a[0]. After transforming a into a BinaryHeap, the heap- sort algorithm repeatedly swaps a[0] and a[n − 1], decrements n, and calls trickleDown(0) so that a[0], . . . , a[n − 2] once again are a valid heap representation. When this process ends (because n = 0) the elements of a are stored in decreasing order, so a is reversed to obtain the final sorted order.1 Figure 11.1.3 shows an example of the execution of heapSort(a, c). BinaryHeap void sort(array<T> &b) { BinaryHeap<T> h(b); while (h.n > 1) { h.a.swap(--h.n, 0); h.trickleDown(0); } b = h.a; b.reverse(); } 1 The algorithm could alternatively redefine the compare(x, y) function so that the heap sort algorithm stores the elements directly in ascending order. 180 11. Sorting Algorithms 11.1. Comparison-Based Sorting A key subroutine in heap sort is the constructor for turning an unsorted array a into a heap. It would be easy to do this in O(n log n) time by repeatedly calling the BinaryHeap add(x) method, but we can do better by using a bottom-up algorithm. Recall that, in a binary heap, the children of a[i] are stored at positions a[2i + 1] and a[2i + 2]. This implies that the elements a[bn/2c], . . . , a[n − 1] have no children. In other words, each of a[bn/2c], . . . , a[n − 1] is a sub-heap of size 1. Now, working backwards, we can call trickleDown(i) for each i ∈ {bn/2c − 1, . . . , 0}. This works, because by the time we call trickleDown(i), each of the two children of a[i] are the root of a sub-heap so calling trickleDown(i) makes a[i] into the root of its own subheap. BinaryHeap BinaryHeap(array<T> &b) : a(0) { a = b; n = a.length; for (int i = n/2-1; i >= 0; i--) { trickleDown(i); } } The interesting thing about this bottom-up strategy is that it is more efficient than calling add(x) n times. To see this, notice that, for n/2 elements, we do no work at all, for n/4 elements, we call trickleDown(i) on a subheap rooted at a[i] and whose height is 1, for n/8 elements, we call trickleDown(i) on a subheap whose height is 2, and so on. Since the work done by trickleDown(i) is proportional to the height of the sub-heap rooted at a[i], this means that the total work done is at most Xn log ∞ X ∞ X i i O((i − 1)n/2 ) ≤ O(in/2 ) = O(n) i/2i = O(2n) = O(n) . i=1 i=1 i=1 P∞ i The second-last equality follows by recognizing that the sum i=1 i/2 is equal, by defini- tion, to the expected number times we toss a coin up to and including the first time the coin comes up as heads and applying Lemma 4.2. The following theorem describes the performance of heapSort(a, c). Theorem 11.4. The heapSort(a, c) method runs in O(n log n) time and performs at most 2n log n + O(n) comparisons. Proof. The algorithm runs in 3 steps: (1) Transforming a into a heap, (2) repeatedly extracting the minimum element from a, and (3) reversing the elements in a. We have just argued that step 1 takes O(n) time and performs O(n) comparisons. Step 3 takes O(n) 181 11. Sorting Algorithms 11.1. Comparison-Based Sorting time and performs no comparisons. Step 2 performs n calls to trickleDown(0). The ith such call operates on a heap of size n − i and performs at most 2 log(n − i) comparisons. Summing this over i gives n−i X n−i X 2 log(n − i) ≤ 2 log n = 2n log n i=0 i=0 Adding the number of comparisons performed in each of the three steps completes the proof. 11.1.4 A Lower-Bound for Comparison-Based Sorting We have now seen three comparison-based sorting algorithms that each run in O(n log n) time. By now, we should be wondering if faster algorithms exist. The short answer to this question is no. If the only operations allowed on the elements of a are comparisons then no algorithm can avoid doing roughly n log n comparisons. This is not difficult to prove, but requires a little imagination. Ultimately, it follows from the fact that log(n!) = log n + log(n − 1) + · · · + log(1) = n log n − O(n) . (Proving this fact is left as Exercise 11.4.) We will first focus our attention on deterministic algorithms like merge-sort and heap-sort and on a particular fixed value of n. Imagine such an algorithm is being used to sort n distinct elements. The key to proving the lower-bound is to observe that, for a deterministic algorithm with a fixed value of n, the first pair of elements that are com- pared is always the same. For example, in heapSort(a, c), when n is even, the first call to trickleDown(i) is with i = n/2 − 1 and the first comparison is between elements a[n/2 − 1] and a[n − 1]. Since all input elements are distinct, this first comparison has only two possible outcomes. The second comparison done by the algorithm may depend on the outcome of the first comparison. The third comparison may depend on the results of the first two, and so on. In this way, any deterministic comparison-based sorting algorithm can be viewed as a rooted binary comparison-tree. Each internal node, u, of this tree is labelled with a pair of of indices u.i and u.j. If a[u.i] < a[u.j] the algorithm proceeds to the left subtree, otherwise it proceeds to the right subtree. Each leaf w of this tree is labelled with a permutation w.p[0], . . . , w.p[n − 1] of 0, . . . , n − 1. This permutation represents the permutation that is required to sort a if the comparison tree reaches this leaf. That is, a[w.p[0]] < a[w.p[1]] < · · · < a[w.p[n − 1]] . 182 11. Sorting Algorithms 11.1. Comparison-Based Sorting a[0] ≶ a[1] < > a[1] ≶ a[2] a[0] ≶ a[2] < > < > a[0] < a[1] < a[2] a[0] ≶ a[2] a[1] < a[0] < a[2] a[1] ≶ a[2] < > < > a[0] < a[2] < a[1] a[2] < a[0] < a[1] a[1] < a[2] < a[0] a[2] < a[1] < a[0] Figure 11.5: A comparison tree for sorting an array a[0], a[1], a[2] of length n = 3. a[0] ≶ a[1] < > a[1] ≶ a[2] a[0] ≶ a[2] < > < > a[0] < a[1] < a[2] a[0] < a[2] < a[1] a[1] < a[0] < a[2] a[1] < a[2] < a[0] Figure 11.6: A comparison tree that does not correctly sort every input permutation. An example of a comparison tree for an array of size n = 3 is shown in Figure 11.5. The comparison tree for a sorting algorithm tells us everything about the algorithm. It tells us exactly the sequence of comparisons that will be performed for any input array a having n distinct elements and it tells us how the algorithm will reorder a to sort it. An immediate consequence of this is that the comparison tree must have at least n! leaves; if not, then there are two distinct permutations that lead to the same leaf, so the algorithm does not correctly sort at least one of these permutations. For example, the comparison tree in Figure 11.6 has only 4 < 3! = 6 leaves. Inspect- ing this tree, we see that the two input arrays 3, 1, 2 and 3, 2, 1 both lead to the rightmost leaf. On the input 3, 1, 2 this leaf correctly outputs a[1] = 1, a[2] = 2, a[0] = 3. However, on the input 3, 2, 1, this node incorrectly outputs a[1] = 2, a[2] = 1, a[0] = 3. This discussion leads to the primary lower-bound for comparison-based algorithms. Theorem 11.5. For any deterministic comparison-based sorting algorithm A and any integer n ≥ 1, there exists an input array a of length n such that A performs at least log(n!) = n log n − O(n) comparisons when sorting a. Proof. By the above discussion, the comparison tree defined by A must have at least n! 183 11. Sorting Algorithms 11.1. Comparison-Based Sorting leaves. An easy inductive proof shows that any binary tree with k leaves has height at least log k. Therefore, the comparison tree for A has a leaf, w, of depth at least log(n!) and there is an input array a that leads to this leaf. The input array a is an input for which A does at least log(n!) comparisons. Theorem 11.5 deals with deterministic algorithms like merge-sort and heap-sort, but doesn’t tell us anything about randomized algorithms like quicksort. Could a randomized algorithm beat the log(n!) lower bound on the number of comparisons? The answer, again, is no. Again, the way to prove it is to think differently about what a randomized algorithm is. In the following discussion, we will implicitly assume that our decision trees have been “cleaned up” in the following way: Any node that can not be reached by some input array a is removed. This cleaning up implies that the tree has exactly n! leaves. It has at least n! leaves because, otherwise, it could not sort correctly. It has at most n! leaves since each of the possible n! permutation of n distinct elements follows exactly one root to leaf path in the decision tree. We can think of a randomized sorting algorithm R as a deterministic algorithm that takes two inputs: The input array a that should be sorted and a long sequence b = b1 , b2 , b3 , . . . , bm of random real numbers in the range [0, 1]. The random numbers provide the randomization. When the algorithm wants to toss a coin or make a random choice, it does so by using some element from b. For example, to compute the index of the first pivot in quicksort, the algorithm could use the formula bnb1 c. Now, notice that if we fix b to some particular sequence b̂ then R becomes a de- terministic sorting algorithm, R(b̂), that has an associated comparison tree, T (b̂). Next, notice that if we select a to be a random permutation of {1, . . . , n}, then this is equivalent to selecting a random leaf, w, from the n! leaves of T (b̂). Exercise 11.6 asks you to prove that, if we select a random leaf from any binary tree with k leaves, then the expected depth of that leaf is at least log k. Therefore, the expected number of comparisons performed by the (deterministic) algorithm R(b̂) when given an input array containing a random permutation of {1, . . . , n} is at least log(n!). Finally, notice that this is true for every choice of b̂, therefore it holds even for R. This completes the proof of the lower-bound for randomized algorithms. Theorem 11.6. For any (deterministic or randomized) comparison-based sorting algorithm A and any integer n ≥ 1, the expected number of comparisons done by A when sorting a 184 11. Sorting Algorithms 11.2. Counting Sort and Radix Sort random permutation of {1, . . . , n} is at least log(n!) = n log n − O(n). 11.2 Counting Sort and Radix Sort In this section we consider two sorting algorithms that are not comparison-based. These algorithms are specialized for sorting small integers. These algorithms get around the lower-bounds of Theorem 11.5 by using (parts of) the elements of a as indices into an array. Consider a statement of the form c[a[i]] = 1 . This statement executes in constant time, but has c.length possible different outcomes, depending on the value of a[i]. This means that the execution of an algorithm that makes such a statement can not be modelled as a binary tree. Ultimately, this is the reason that the algorithms in this section are able to sort faster than comparison-based algorithms. 11.2.1 Counting Sort Suppose we have an input array a consisting of n integers, each in the range 0, . . . , k − 1. The counting-sort algorithm sorts a using an auxiliary array c of counters. It outputs a sorted version of a as an auxiliary array b. The idea behind counting-sort is simple: For each i ∈ {0, . . . , k − 1}, count the number of occurrences of i in a and store this in c[i]. Now, after sorting, the output will look like c[0] occurrences of 0, followed by c[1] occurrences of 1, followed by c[2] occurrences of 2,. . . , followed by c[k − 1] occurrences of k − 1. The code that does this is very slick, and its execution is illustrated in Figure 11.7: Algorithms void countingSort(array<int> &a, int k) { array<int> c(k, 0); for (int i = 0; i < a.length; i++) c[a[i]]++; for (int i = 1; i < k; i++) c[i] += c[i-1]; array<int> b(a.length); for (int i = a.length-1; i >= 0; i--) b[--c[a[i]]] = a[i]; a = b; } The first for loop in this code sets each counter c[i] so that it counts the number of occurrences of i in a. By using the values of a as indices, these counters can all be computed in O(n) time with a single for loop. At this point, we could use c to fill in the 185 11. Sorting Algorithms 11.2. Counting Sort and Radix Sort a 7 2 9 0 1 2 0 9 7 4 4 6 9 1 0 9 3 2 5 9 c 3 2 3 1 2 1 1 2 0 5 0 1 2 3 4 5 6 7 8 9 0 c 3 5 8 9 11 12 13 15 15 20 b 0 0 0 1 1 2 2 2 3 4 4 5 6 7 7 9 9 9 9 9 0 1 2 3 4 5 6 8 7 9 0 c 3 5 8 9 11 12 13 15 20 a 7 2 9 0 1 2 0 9 7 4 4 6 9 1 0 9 3 2 5 9 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 11.7: The operation of counting sort on an array of length n = 20 that stores integers 0, . . . , k − 1 = 9. output array b directly. However, this would not work if the elements of a have associated data. Therefore we spend a little extra effort to copy the elements of a into b. The next for loop, which takes O(k) time, computes a running-sum of the counters so that c[i] becomes the number of elements in a that are less than or equal to i. In particular, for every i ∈ {0, . . . , k − 1}, the output array, b, will have b[c[i − 1]] = b[c[i − 1] + 1] = · · · = b[c[i] − 1] = i . Finally, the algorithm scans a backwards to put its elements in order into an output array b. When scanning, the element a[i] = j is placed at location b[c[j] − 1] and the value c[j] is decremented. Theorem 11.7. The countingSort(a, k) method can sort an array a containing n integers in the set {0, . . . , k − 1} in O(n + k) time. The counting-sort algorithm has the nice property of being stable; it preserves the relative order of elements that are equal. If two elements a[i] and a[j] have the same value, and i < j then a[i] will appear before a[j] in b. This will be useful in the next section. 186 11. Sorting Algorithms 11.2. Counting Sort and Radix Sort 11.2.2 Radix-Sort Counting-sort is very efficient for sorting an array of integers when the length, n, of the array is not much smaller than the maximum value, k − 1, that appears in the array. The radix-sort algorithm, which we now describe, uses several passes of counting-sort to allow for a much greater range of maximum values. Radix-sort sorts w-bit integers by using w/d passes of counting-sort to sort these integers d bits at a time.2 More precisely, radix sort first sorts the integers by their least significant d bits, then their next significant d bits, and so on until, in the last pass, the integers are sorted by their most significant d bits. Algorithms void radixSort(array<int> &a) { int d = 8, w = 32; for (int p = 0; p < w/d; p++) { array<int> c(1<<d, 0); // the next three for loops implement counting-sort array<int> b(a.length); for (int i = 0; i < a.length; i++) c[(a[i] >> d*p)&((1<<d)-1)]++; for (int i = 1; i < 1<<d; i++) c[i] += c[i-1]; for (int i = a.length-1; i >= 0; i--) b[--c[(a[i] >> d*p)&((1<<d)-1)]] = a[i]; a = b; } } (In this code, the expression (a[i] >> d ∗ p)&((1 << d) − 1) extracts the integer whose binary representation is given by bits (p + 1)d − 1, . . . , pd of a[i].) An example of the steps of this algorithm is shown in Figure 11.8. This remarkable algorithm sorts correctly because counting-sort is a stable sorting algorithm. If x < y are two elements of a and the most significant bit at which x differs from y has index r, then x will be placed before y during pass br/dc and subsequent passes will not change the relative order of x and y. Radix-sort performs w/d passes of counting-sort. Each pass requires O(n + 2d ) time. Therefore, the performance of radix-sort is given by the following theorem. Theorem 11.8. For any integer d > 0, the radixSort(a, k) method can sort an array a containing n w-bit integers in O((w/d)(n + 2d )) time. 2 We assume that d divides w, otherwise we can always increase w to ddw/de. 187 11. Sorting Algorithms 11.3. Discussion and Exercises 01010001 11001000 11110000 00000001 00000001 00000001 00101000 01010001 11001000 00001111 11001000 11110000 00000001 00001111 00101000 00101000 01010001 01010101 01010001 01010001 00001111 00000001 11001000 01010101 01010101 11110000 01010101 00101000 00101000 10101010 10101010 10101010 10101010 10101010 11001000 01010101 00001111 00001111 11110000 11110000 Figure 11.8: Using radixsort to sort w = 8-bit integers by using 4 passes of counting sort on d = 2-bit integers. If we think, instead, of the elements of the array being in the range {0, . . . , nc − 1}, and take d = dlog ne we obtain the following version of Theorem 11.8. Corollary 11.1. The radixSort(a, k) method can sort an array a containing n integer values in the range {0, . . . , nc − 1} in O(cn) time. 11.3 Discussion and Exercises Sorting is probably the fundamental algorithmic problem in computer science, and has a long history. Knuth [41] attributes the merge-sort algorithm to von Neumann (1945). Quicksort is due to Hoare [33]. The original heap-sort algorithm is due to Williams [63], but the version presented here (in which the heap is constructed bottom-up in O(n) time) is due to Floyd [23]. Lower-bounds for comparison-based sorting appear to be folklore. The following table summarizes the performance of these comparison-based algorithms: comparisons in-place Merge-sort n log n worst-case No Quicksort 1.38n log n + O(n) expected Yes Heap-sort 2n log n + O(n) worst-case Yes Each of these comparison-based algorithms has advantages and disadvantages. Merge- sort does the fewest comparisons and does not rely on randomization. Unfortunately, it uses an auxilliary array during its merge phase. Allocating this array can be expensive and is a potential point of failure if memory is limited. Quicksort is an in-place algorithm and is a close second in terms of the number of comparisons, but is randomized so this running 188 11. Sorting Algorithms 11.3. Discussion and Exercises time is not always guaranteed. Heap-sort does the most comparisons, but it is in-place and deterministic. There is one setting in which merge-sort is a clear-winner; this occurs when sorting a linked-list. In this case, the auxiliary array is not needed; two sorted linked lists are very easily merged into a single sorted linked-list by pointer manipulations. The counting-sort and radix-sort algorithms described here are due to Seward [57, Section 2.4.6]. However, variants of radix-sort have been used since the 1920’s to sort punch cards using punched card sorting machines. These machines can sort a stack of cards into two piles based on the existence (or not) of a hole in a specific location on the card. Repeating this process for different hole locations gives an implementation of radix-sort. Exercise 11.1. Implement a version of the merge-sort algorithm that sorts a DLList without using an auxiliary array. Exercise 11.2. Analyze the expected number of comparisons done by Quicksort a little more carefully than the proof of Theorem 11.3. In particular, show that the expected number of comparisons is 2nHn − n + Hn . Exercise 11.3. Find another pair of permutations of 1, 2, 3 that are not correctly sorted by the comparison-tree in Figure 11.6. Exercise 11.4. Prove that log n! = n log n − O(n). Exercise 11.5. Prove that a binary tree with k leaves has height at least log k. Exercise 11.6. Prove that, if we pick a random leaf from a binary tree with k leaves, then the expected height of this leaf is at least log k. (Hint: Use induction along with the inequality (k1 /k) log k1 + (k2 /k) log k2 ) ≥ log k − 1, when k1 + k2 = k.) Exercise 11.7. Simple floating point numbers are numbers of the form x · 10y , where 0 ≤ x ≤ 1 and y is an integer and each of x and y can be represented by at most k bits. Describe a version of radix-sort that can be used to sort simple floating point numbers. Extend your version of radix sort so that it can handle signed values of both x and y and implement a version of radix-sort that sorts arrays of type float. 189 11. Sorting Algorithms 11.3. Discussion and Exercises 190 Chapter 12 Graphs In this chapter, we study two representations of graphs and basic algorithms on these representations. Mathematically, a (directed) graph is a pair G = (V, E) where V is a set of vertices and E is a set of ordered pairs of vertices called edges. An edge (i, j) is directed from i to j; i is called the source of the edge and j is called the target. A path in G is a sequence of vertices v0 , . . . , vk such that, for every i ∈ {1, . . . , k}, the edge (vi−1 , vi ) is in E. A path v0 , . . . , vk is a cycle if, additionally, the edge (vk , v0 ) is in E. A path (or cycle) is simple if all of its vertices are unique. If there is a path from some vertex vi to some vertex vj then we say that vj is reachable from vi . An example of a graph is shown in Figure 12.1. Graphs have an enormous number of applications, due to their ability to model so many phenomenon. There are many obvious examples. Computer networks can be modelled as graphs, with vertices corresponding to computers and edges corresponding to (directed) communication links between those computers. Street networks can be modelled as graphs, with vertices representing intersections and edges representing streets joining consecutive intersections. Less obvious examples occur as soon as we realize that graphs can model any pairwise relationships within a set. For example, in a university setting we might have a timetable conflict graph whose vertices represent courses offered in the university and in which the edge (i, j) is present if and only if there is at least one student that is taking both class i and class j. Thus, an edge indicates that the exam for class i can not be scheduled at the same time as the exam for class j. Throughout this section, we will use n to denote the number of vertices of G and m to denote the number of edges of G. That is, n = |V | and m = |E|. Furthermore, we will assume that V = {0, . . . , n − 1}. Any other data that we would like to associate with the 191 12. Graphs 0 1 2 3 4 5 6 7 8 9 10 11 Figure 12.1: A graph with 12 vertices. Vertices are drawn as numbered circles and edges are drawn as pointed curves pointing from source to target. elements of V can be stored in an array of length n. Some typical operations performed on graphs are: • addEdge(i, j): Add the edge (i, j) to E. • removeEdge(i, j): Remove the edge (i, j) from E. • hasEdge(i, j): Check if the edge (i, j) ∈ E • outEdges(i): Return a List of all integers j such that (i, j) ∈ E • inEdges(i): Return a List of all integers j such that (j, i) ∈ E Note that these operations are not terribly difficult to implement efficiently. For example, the first three operations can be implemented directly by using a USet, so they can be implemented in constant expected time using the hash tables discussed in Chapter 5. The last two operations can be implemented in constant time by storing, for each vertex, a list of its adjacent vertices. However, different applications of graphs have different performance requirements for these operations and, ideally, we can use the simplest implementation that satisfies all the application’s requirements. For this reason, we discuss two broad categories of graph representations. 192 12. Graphs 12.1. AdjacencyMatrix: Representing a Graph by a Matrix 12.1 AdjacencyMatrix: Representing a Graph by a Matrix An adjacency matrix is a way of representing an n vertex graph G = (V, E) by an n × n matrix, a, whose entries are boolean values. AdjacencyMatrix int n; bool **a; The matrix entry a[i][j] is defined as true if (i, j) ∈ E a[i][j] = false otherwise The adjacency matrix for the graph in Figure 12.1 is shown in Figure 12.2. With this representation, the addEdge(i, j), removeEdge(i, j), and hasEdge(i, j) operations just involve setting or reading the matrix entry a[i][j]: AdjacencyMatrix void addEdge(int i, int j) { a[i][j] = true; } void removeEdge(int i, int j) { a[i][j] = false; } bool hasEdge(int i, int j) { return a[i][j]; } These operations clearly take constant time per operation. Where the adjacency matrix performs poorly is with the outEdges(i) and inEdges(i) operations. To implement these, we must scan all n entries in the corresponding row or column of a and gather up all the indices, j, where a[i][j], respectively a[j][i], is true. AdjacencyMatrix void outEdges(int i, List &edges) { for (int j = 0; j < n; j++) if (a[i][j]) edges.add(j); } void inEdges(int i, List &edges) { for (int j = 0; j < n; j++) if (a[j][i]) edges.add(j); } 193 12. Graphs 12.1. AdjacencyMatrix: Representing a Graph by a Matrix 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 0 0 0 2 1 0 0 1 0 0 1 0 0 0 0 0 3 0 0 1 0 0 0 0 1 0 0 0 0 4 1 0 0 0 0 1 0 0 1 0 0 0 5 0 1 1 0 1 0 1 0 0 1 0 0 6 0 0 1 0 0 1 0 1 0 0 1 0 7 0 0 0 1 0 0 1 0 0 0 0 1 8 0 0 0 0 1 0 0 0 0 1 0 0 9 0 0 0 0 0 1 0 0 1 0 1 0 10 0 0 0 0 0 0 1 0 0 1 0 1 11 0 0 0 0 0 0 0 1 0 0 1 0 Figure 12.2: A graph and its adjacency matrix. 194 12. Graphs 12.2. AdjacencyLists: A Graph as a Collection of Lists These operations clearly take O(n) time per operation. Another drawback of the adjacency matrix representation is that it is big. It stores an n × n boolean matrix, so it requires at least n2 bits of memory. The implementation here uses a matrix of bool values so it actually uses on the order of n2 bytes of memory. A more careful implementation, that packs w boolean values into each word of memory could reduce this space usage to O(n2 /w) words of memory. Theorem 12.1. The AdjacencyMatrix data structure implements the Graph interface. An AdjacencyMatrix supports the operations • addEdge(i, j), removeEdge(i, j), and hasEdge(i, j) in constant time per operation; and • inEdges(i), and outEdges(i) in O(n) time per operation. The space used by an AdjacencyMatrix is O(n2 ). Despite the high memory usage and poor performance of the inEdges(i) and outEdges(i) operations, an AdjacencyMatrix can still be useful for some applications. In particular, when the graph G is dense, i.e., it has close to n2 edges, then a memory usage of n2 may be acceptable. The AdjacencyMatrix data structure is also commonly used because algebraic op- erations the matrix a can be used to efficiently compute properties of the graph G. This is a topic for a course on algorithms, but we point out one such property here: If we treat the entries of a as integers (1 for true and 0 for false) and multiply a by itself using matrix multiplication then we get the matrix a2 . Recall, from the definition of matrix multiplication, that n−1 X a2 [i][j] = a[i][k] · a[k][j] . k=0 Interpreting this sum in terms of the graph G, this formula counts the number of vertices, k, such that G contains both edges (i, k) and (k, j). That is, it counts the number of paths from i to j (through intermediate vertices, k) that have length exactly 2. This observation is the foundation of an algorithm that computes the shortest paths between all pairs of vertices in G using only O(log n) matrix multiplications. 12.2 AdjacencyLists: A Graph as a Collection of Lists Adjacency list representations takes a more vertex-centric approach. There are many differ- ent possible implementations of adjacency lists. In this section, we present a simple one. At 195 12. Graphs 12.2. AdjacencyLists: A Graph as a Collection of Lists 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 1 0 1 2 0 1 5 6 4 8 9 10 4 2 3 7 5 2 2 3 9 5 6 7 6 6 8 6 7 11 10 11 5 9 10 4 Figure 12.3: A graph and its adjacency lists the end of the section, we discuss different possibilities. In an adjacency list representation, the graph G = (V, E) is represented as an array, adj, of lists. The list adj[i] contains a list of all the vertices adjacent to vertex i. That is, it contains every index j such that (i, j) ∈ E. AdjacencyLists int n; List *adj; (An example is shown in Figure 12.3.) In this particular implementation, we represent each list in adj as a subclass of ArrayStack, because we would like constant time access by position. Other options are also possible. Specifically, we could have implemented adj as a DLList. The addEdge(i, j) operation just appends the value j to the list adj[i]: AdjacencyLists void addEdge(int i, int j) { adj[i].add(j); } This takes constant time. 196 12. Graphs 12.2. AdjacencyLists: A Graph as a Collection of Lists The removeEdge(i, j) operation searches through the list adj[i] until it finds j and then removes it: AdjacencyLists void removeEdge(int i, int j) { for (int k = 0; k < adj[i].size(); k++) { if (adj[i].get(k) == j) { adj[i].remove(k); return; } } } This takes O(deg(i)) time, where deg(i) (the degree of i) counts the number of edges in E that have i as their source. The hasEdge(i, j) operation is similar; it searches through the list adj[i] until it finds j (and returns true), or reaches the end of the list (and returns false): AdjacencyLists bool hasEdge(int i, int j) { return adj[i].contains(j); } This also takes O(deg(i)) time. The outEdges(i) operation is very simple; It simply copies the values in adj[i] into the output list: AdjacencyLists void outEdges(int i, LisT &edges) { for (int k = 0; k < adj[i].size(); k++) edges.add(adj[i].get(k)); } This clearly takes O(deg(i)) time. The inEdges(i) operation is much more work. It scans over every vertex j checking if the edge (i, j) exists and, if so, adding j to the output list: AdjacencyLists void inEdges(int i, LisT &edges) { for (int j = 0; j < n; j++) if (adj[j].contains(i)) edges.add(j); } This operation is very slow. It scans the adjacency list of every vertex, so it takes O(n + m) time. 197 12. Graphs 12.3. Graph Traversal The following theorem summarizes the performance of the above data structure: Theorem 12.2. The AdjacencyLists data structure implements the Graph interface. An AdjacencyLists supports the operations • addEdge(i, j) in constant time per operation; • removeEdge(i, j) and hasEdge(i, j) in O(deg(i)) time per operation; • outEdges(i) in O(deg(i)) time per operation; and • inEdges(i) in O(n + m) time per operation. The space used by a AdjacencyLists is O(n + m). As alluded to earlier, there are many different choices to be made when implementing a graph as an adjacency list. Some questions that come up include: • What type of collection should be used to store each element of adj? One could use an array-based list, a linked-list, or even a hashtable. • Should there be a second adjacency list, inadj, that stores, for each i, the list of vertices, j, such that (j, i) ∈ E? This can greatly reduce the running-time of the inEdges(i) operation, but requires slightly more work in the addEdge(i, j) and removeEdge(i, j) operations. • Should the entry for the edge (i, j) in adj[i] be linked by a reference to the corre- sponding entry in inadj[j]. • Should edges be first-class objects with their own associated data? In this way, adj would contain lists of edges rather than lists of vertices (integers). Most of these questions come down to a tradeoff between complexity (and space) of imple- mentation and performance features of the implementation. 12.3 Graph Traversal In this section we present two algorithms for exploring a graph, starting at one of its vertices, i, and finding all vertices that are reachable from i. Both of these algorithms are best suited to graphs represented using an adjacency list representation. Therefore, when analyzing these algorithms we will assume that the underlying representation is as an AdjacencyLists. 198 12. Graphs 12.3. Graph Traversal 12.3.1 Breadth-First Search The bread-first-search algorithm starts at a vertex i and visits, first the neighbours of i, then the neighbours of the neighbours of i, then the neighbours of the neighbours of the neighbours of i, and so on. This algorithm is a generalization of the breadth-first-search algorithm for binary trees (Section 6.1.2), and is very similar; it uses a queue, q, that initially contains only i. It then repeatedly extracts an element from q and adds its neighbours to q, provided that these neighbours have never been in q before. The only major difference between breadth-first-search for graphs and for trees is that the algorithm for graphs has to ensure that it does not add the same vertex to q more than once. It does this by using an auxiliary boolean array, seen, that keeps track of which vertices have already been discovered. Algorithms void bfs(Graph &g, int r) { bool *seen = new bool[g.nVertices()]; SLList<int> q; q.add(r); seen[r] = true; while (q.size() > 0) { int i = q.remove(); ArrayStack<int> edges; g.outEdges(i, edges); for (int k = 0; k < edges.size(); k++) { int j = edges.get(k); if (!seen[j]) { q.add(j); seen[j] = true; } } } delete[] seen; } An example of running bfs(g, 0) on the graph from Figure 12.1 is shown in Figure 12.4. Dif- ferent executions are possible, depending on the ordering of the adjacency lists; Figure 12.4 uses the adjacency lists in Figure 12.3. Analyzing the running-time of the bfs(g, i) routine is fairly straightforward. The use of the seen array ensures that no vertex is added to q more than once. Adding (and later removing) each vertex from q takes constant time per vertex for a total of O(n) time. Since each vertex is processed at most once by the inner loop, each adjacency list is processed at 199 12. Graphs 12.3. Graph Traversal 0 1 3 7 2 5 4 8 6 10 9 11 Figure 12.4: An example of bread-first-search starting at node 0. Nodes are labelled with the order in which they are added to q. Edges that result in nodes being added to q are drawn in black, other edges are drawn in grey. most once, so each edge of G is processed at most once. This processing, which is done in the inner loop takes constant time per iteration, for a total of O(m) time. Therefore, the entire algorithm runs in O(n + m) time. The following theorem summarizes the performance of the bfs(g, r) algorithm. Theorem 12.3. When given as input a Graph, g, that is implemented using the AdjacencyLists data structure, the bfs(g, r) algorithm runs in O(n + m) time. A breadth-first traversal has some very special properties. Calling bfs(g, r) will eventually enqueue (and eventually dequeue) every vertex j such that there is a directed path from r to j. Moreover, the vertices at distance 0 from r (r itself) will enter q before the vertices at distance 1, which will enter q before the vertices at distance 2, and so on. Thus, the bfs(g, r) method visits vertices in increasing order of distance from r and vertices that can not be reached from r are never output at all. A particularly useful application of the breadth-first-search algorithm is, therefore, in computing shortest paths. To compute the shortest path from r to every other vertex, we use a variant of bfs(g, r) that uses an auxilliary array, p, of length n. When a new vertex j is added to q, we set p[j] = i. In this way, p[j] becomes the second last node on a shortest path from r to j. Repeating this, by taking p[p[j], p[p[p[j]]], and so on we can reconstruct the (reversal of) a shortest path from r to j. 200 12. Graphs 12.3. Graph Traversal 12.3.2 Depth-First Search The depth-first-search algorithm is similar to the standard algorithm for traversing binary trees; it first fully explores one subtree before returning to the current node and then exploring the other subtree. Another way to think of depth-first-search is by saying that it is similar to breadth-first search except that it uses a stack instead of a queue. During the execution of the depth-first-search algorithm, each vertex, i, is assigned a color, c[i]: white if we have never seen the vertex before, grey if we are currently visiting that vertex, and black if we are done visiting that vertex. The easiest way to think of depth-first-search is as a recursive algorithm. It starts by visiting r. When visiting a vertex i, we first mark i as grey. Next, we scan i’s adjacency list and recursively visit any white vertex we find in this list. Finally, we are done processing i, so we color i black and return. Algorithms void dfs(Graph &g, int i, char *c) { c[i] = grey; // currently visiting i ArrayStack<int> edges; g.outEdges(i, edges); for (int k = 0; k < edges.size(); k++) { int j = edges.get(k); if (c[j] == white) { c[j] = grey; dfs(g, j, c); } } c[i] = black; // done visiting i } void dfs(Graph &g, int r) { char *c = new char[g.nVertices()]; dfs(g, r, c); delete[] c; } An example of the execution of this algorithm is shown in Figure 12.5 Although depth-first-search may best be thought of as a recursive algorithm, recur- sion is not the best way to implement it. Indeed, the code given above will fail for many large graphs by causing a stack overflow. An alternative implementation is to replace the recursion stack with an explicit stack, s. The following implementation does just that: Algorithms void dfs2(Graph &g, int r) { char *c = new char[g.nVertices()]; SLList<int> s; 201 12. Graphs 12.3. Graph Traversal 0 1 2 3 9 10 11 4 8 7 6 5 Figure 12.5: An example of depth-first-search starting at node 0. Nodes are labelled with the order in which they are processed. Edges that result in a recursive call are drawn in black, other edges are drawn in grey. s.push(r); while (s.size() > 0) { int i = s.pop(); if (c[i] == white) { c[i] = grey; ArrayStack<int> edges; g.outEdges(i, edges); for (int k = 0; k < edges.size(); k++) s.push(edges.get(k)); } } delete[] c; } In the above code, when next vertex, i, is processed, i is colored grey and then replaced, on the stack, with its adjacent vertices. During the next iteration, one of these vertices will be visited. Not surprisingly, the running times of dfs(g, r) and dfs2(g, r) are the same as that of bfs(g, r): Theorem 12.4. When given as input a Graph, g, that is implemented using the AdjacencyLists data structure, the dfs(g, r) and dfs2(g, r) algorithms each run in O(n + m) time. As with the breadth-first-search algorithm, there is an underlying tree associated with each execution of depth-first-search. When a node i 6= r goes from white to grey, this 202 12. Graphs 12.4. Discussion and Exercises P j i C Figure 12.6: The depth-first-search algorithm can be used to detect cycles in G.The node j is colored grey while i is still grey. This implies there is a path, P , from i to j in the depth-first-search tree, and the edge (j, i) implies that P is also a cycle. is because dfs(g, i, c) was called recursively while processing some node i0 . (In the case of dfs2(g, r) algorithm, i is one of the nodes that replaced i0 on the stack.) If we think of i0 as the parent of i, then we obtain a tree rooted at r. In Figure 12.5, this tree is a path from vertex 0 to vertex 11. An important property of the depth-first-search algorithm is the following: Suppose that when node i is colored grey, there exists a path from i to some other node j that uses only white vertices. Then j will be colored (first grey then) black before i is colored black. (This can be proven by contradiction, by considering any path P from i to j.) One application of this property is the detection of cycles. Refer to Figure 12.6. Consider some cycle, C, that can be reached from r. Let i be the first node of C that is colored grey, and let j be the node that precedes i on the cycle C. Then, by the above property, j will be colored grey and the edge (j, i) will be considered by the algorithm while i is still grey. Thus, the algorithm can conclude that there is a path, P , from i to j in the depth-first-search tree and the edge (j, i) exists. Therefore, P is also a cycle. 12.4 Discussion and Exercises The running times of the depth-first-search and breadth-first-search algorithms are some- what overstated by the Theorems 12.3 and 12.4. Define nr as the number of vertices, i, of G, for which there exists a path from r to i. Define mr as the number of edges that have these vertices as their sources. Then the following theorem is a more precise statement of the running times of the breadth-first-search and depth-first-search algorithms. (This more refined statement of the running time is useful in some of the applications of these algorithms outlined in the exercises.) Theorem 12.5. When given as input a Graph, g, that is implemented using the AdjacencyLists data structure, the bfs(g, r), dfs(g, r) and dfs2(g, r) algorithms each run in O(nr + mr ) 203 12. Graphs 12.4. Discussion and Exercises time. Breadth-first search seems to have been discovered independently by Moore [43] and Lee [42] in the contexts of maze exploration and circuit routing, respectively. Adjacency-list representations of graphs were first popularized by Hopcroft and Tar- jan [34] as an alternative to the (then more common) adjacency-matrix representation. This representation, and depth-first-search, played a major part in the celebrated Hopcroft- Tarjan planarity testing algorithm that can determine, in O(n) time, if a graph can be drawn, in the plane, and in such a way that no pair of edges cross each other [35]. In the following exercises, an undirected graph is one in which, for every i and j, the edge (i, j) is present if and only if the edge (j, i) is present. Exercise 12.1. Let G be an undirected graph. We say G is connected if, for every pair of vertices i and j in G, there is a path from i to j. Show how to test if G is connected in O(n + m) time. Exercise 12.2. We say G is connected if, for every pair of vertices i and j in G, there is a path from i to j. Show how to test if G is connected in O(n + m) time. Exercise 12.3. Let G be an undirected graph. A connected-component labelling of G par- titions to the vertices of G into maximal sets, each of which forms a connected subgraph. Show how to compute a connected component labelling of G in O(n + m) time. Exercise 12.4. Let G be an undirected graph. A spanning forest of G is a collection of trees, one per component, whose edges are edges of G and whose vertices contain all vertices of G. Show how to compute a spanning forest of of G in O(n + m) time. Exercise 12.5. Given a graph G = (V, E) and some special vertex r ∈ V , show how to compute the length of the shortest path from r to i for every vertex i ∈ V . Exercise 12.6. Give a (simple) example where the dfs(g, r) code visits the nodes of a graph in an order that is different from that of the dfs2(g, r) code. Write a version of dfs2(g, r) that always visits nodes in exactly the same order as dfs(g, r). (Hint: Just start tracing the execution of each algorithm on some graph where r is the source of more than 1 edge.) 204 Chapter 13 Data Structures for Integers In this chapter, we return to the problem of implementing an SSet. The difference now is that we assume the elements stored in the SSet are w-bit integers. That is, we want to implement add(x), remove(x), and find(x) where x ∈ {0, . . . , 2w − 1}. It is not too hard to think of plenty of applications where the data—or at least the key that we use for sorting the data—is an integer. We will discuss three data structures, each building on the ideas of the previous. The first structure, the BinaryTrie performs all three SSet operations in O(w) time. This is not very impressive, since any subset of {0, . . . , 2w − 1} has size n ≤ 2w , so that log n ≤ w. All the other SSet implementations discussed in this book perform all operations in O(log n) time so they are all at least as fast as a BinaryTrie. The second structure, the XFastTrie, speeds up the search in a BinaryTrie by using hashing. With this speedup, the find(x) operation runs in O(log w) time. However, add(x) and remove(x) operations in an XFastTrie still take O(w) time and the space used by an XFastTrie is O(n · w). The third data structure, the YFastTrie, uses an XFastTrie to store only a sample of roughly one out of every w elements and stores the remaining elements a standard SSet structure. This trick reduces the running time of add(x) and remove(x) to O(log w) and decreases the space to O(n). The implementations used as examples in this chapter can store any type of data, as long an integer can be associated with it. In the code samples, the variable ix is always the integer value associated with x, and the method intValue(x) converts x to its associated integer. In the text, however, we will simply treat x as if it is an integer. 205 13. Data Structures for Integers 13.1. BinaryTrie: A digital search tree ???? 0? ? ? 1? ? ? 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 13.1: The integers stored in a binary trie are encoded as root-to-leaf paths. 13.1 BinaryTrie: A digital search tree A BinaryTrie encode a set of w bit integers in a binary tree. All leaves in the tree have depth w and each integer is encoded as a root-to-leaf path. The path for the integer x turns left at level i if the ith most significant bit of x is a 0 and turns right if it is a 1. Figure 13.1 shows an example for the case w = 4, in which the trie stores the integers 3(0011), 9(1001), 12(1100), and 13(1101). Because the search path for a value x depends on the bits of x it will be helpful to name the children of a node, u, u.child[0] (left) and u.child[1] (right). These child pointers will actually serve double-duty. Since the leaves in a binary trie have no children, the pointers are used to string the leaves together into a doubly-linked list. For a leaf in the binary trie u.child[0] (prev) is the node that comes before u in the list and u.child[1] (next) is the node that follows u in the list. A special node, dummy, is used both before the first node and after the last node in the list (see Section 3.2). In the code samples, u.child[0], u.left, and u.prev refer to the same field in the node u, as do u.child[1], u.right, and u.next. Each node, u, also contains an additional pointer u.jump. If u’s left child is missing, then u.jump points to the smallest leaf in u’s subtree. If u’s right child is missing, then u.jump points to the largest leaf in u’s subtree. An example of a BinaryTrie, showing jump pointers and the doubly-linked list at the leaves, is shown in Figure 13.2 The find(x) operation in a BinaryTrie is fairly straightforward. We try to follow the search path for x in the trie. If we reach a leaf, then we have found x. If, we reach a 206 13. Data Structures for Integers 13.1. BinaryTrie: A digital search tree ???? 0? ? ? 1? ? ? 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 13.2: A BinaryTrie with jump pointers shown as curved dashed edges. node u where we cannot proceed (because u is missing a child) then we follow u.jump, which takes us either to smallest leaf larger than x or the largest leaf smaller than x. Which of these two cases occurs depends on whether u is missing its left or right child, respectively. In the former case (u is missing its left child), we have found the value we are looking for. In the latter case (u is missing its right child), we can use the linked list to reach the value we are looking for. Each of these cases is illustrated in Figure 13.3. BinaryTrie T find(T x) { int i, c = 0; unsigned ix = intValue(x); Node *u = &r; for (i = 0; i < w; i++) { c = (ix >> (w-i-1)) & 1; if (u->child[c] == NULL) break; u = u->child[c]; } if (i == w) return u->x; // found it u = (c == 0) ? u->jump : u->jump->next; return u == &dummy ? NULL : u->x; } The running-time of the find(x) method is dominated by the time it takes to follow a root-to-leaf path, so it runs in O(w) time. The add(x) operation in a BinaryTrie is fairly straightforward, but it has a lot of things to take care of: 1. It follows the search path for x until reaching a node u where it can no longer proceed. 207 13. Data Structures for Integers 13.1. BinaryTrie: A digital search tree ???? find(5) 0? ? ? 1? ? ? find(8) 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 13.3: The paths followed by find(5) and find(8). ???? 0? ? ? 1? ? ? 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 13.4: Adding the values 2 and 15 to the BinaryTrie in Figure 13.2. 2. It creates the remainder of the search path from u to a leaf that contains x. 3. It adds the node, u0 , containing x to the linked list of leaves (it has access to u0 ’s prede- cessor, pred, in the linked list from the jump pointer of the last node, u, encountered during step 1.) 4. It walks back up the search path for x adjusting jump pointers at the nodes whose jump pointer should now point to x. An addition is illustrated in Figure 13.4. BinaryTrie bool add(T x) { int i, c = 0; 208 13. Data Structures for Integers 13.1. BinaryTrie: A digital search tree unsigned ix = intValue(x); Node *u = &r; // 1 - search for ix until falling out of the trie for (i = 0; i < w; i++) { c = (ix >> (w-i-1)) & 1; if (u->child[c] == NULL) break; u = u->child[c]; } if (i == w) return false; // trie already contains x - abort Node *pred = (c == right) ? u->jump : u->jump->left; // save for step 3 u->jump = NULL; // u will have two children shortly // 2 - add path to ix for (; i < w; i++) { c = (ix >> (w-i-1)) & 1; u->child[c] = new Node(); u->child[c]->parent = u; u = u->child[c]; } u->x = x; // 3 - add u to linked list u->prev = pred; u->next = pred->next;; u->prev->next = u; u->next->prev = u; // 4 - walk back up, updating jump pointers Node *v = u->parent; while (v != NULL) { if ((v->left == NULL && (v->jump == NULL || intValue(v->jump->x) > ix)) || (v->right == NULL && (v->jump == NULL || intValue(v->jump->x) < ix))) v->jump = u; v = v->parent; } n++; return true; } This method performs one walk down the search path for x and one walk back up. Each step of these walks takes constant time, so the add(x) runs in O(w) time. The remove(x) operation undoes the work of add(x). Like add(x), it has a lot of things to take care of: 1. It follows the search path for x until reaching the leaf, u, containing x. 209 13. Data Structures for Integers 13.1. BinaryTrie: A digital search tree ???? 0? ? ? 1? ? ? 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Figure 13.5: Removing the value 9 from the BinaryTrie in Figure 13.2. 2. It removes u from the doubly-linked list 3. It deletes u and then walks back up the search path for x deleting nodes until reaching a node v that has a child that is not on the search path for x 4. It walks upwards from v to the root updating any jump pointers that point to u. A removal is illustrated in Figure 13.5. BinaryTrie bool remove(T x) { // 1 - find leaf, u, containing x int i = 0, c; unsigned ix = intValue(x); Node *u = &r; for (i = 0; i < w; i++) { c = (ix >> (w-i-1)) & 1; if (u->child[c] == NULL) return false; u = u->child[c]; } // 2 - remove u from linked list u->prev->next = u->next; u->next->prev = u->prev; Node *v = u; // 3 - delete nodes on path to u for (i = w-1; i >= 0; i--) { c = (ix >> (w-i-1)) & 1; v = v->parent; delete v->child[c]; v->child[c] = NULL; 210 13. Data Structures for Integers 13.2. XFastTrie: Searching in Doubly-Logarithmic Time if (v->child[1-c] != NULL) break; } // 4 - update jump pointers v->jump = u; for (; i >= 0; i--) { c = (ix >> (w-i-1)) & 1; if (v->jump == u) v->jump = u->child[1-c]; v = v->parent; } n--; return true; } Theorem 13.1. A BinaryTrie implements the SSet interface for w-bit integers. A BinaryTrie supports the operations add(x), remove(x), and find(x) in O(w) time per operation. The space used by a BinaryTrie that stores n values is O(n · w). 13.2 XFastTrie: Searching in Doubly-Logarithmic Time The performance of the BinaryTrie structure is not that impressive. The number of el- ements, n, stored in the structure is at most 2w , so log n ≤ w. In other words, any of the comparison-based SSet structures described in other parts of this book are at least as efficient as a BinaryTrie, and don’t have the restriction of only being able to store integers. Next we describe the XFastTrie, which is just a BinaryTrie with w + 1 hash tables—one for each level of the trie. These hash tables are used to speed up the find(x) operation to O(log w) time. Recall that the find(x) operation in a BinaryTrie is almost complete once we reach a node, u, where the search path for x would like to proceed to u.right (or u.left) but u has no right (respectively, left) child. At this point, the search uses u.jump to jump to a leaf, v, of the BinaryTrie and either return v or its successor in the linked list of leaves. An XFastTrie speeds up the search process by using binary search on the levels of the trie to locate the node u. To use binary search, we need a way to determine if the node u we are looking for is above a particular level, i, of if u is at or below level i. This information is given by the highest-order i bits in the binary representation of x; these bits determine the search path that x takes from the root to level i. For an example, refer to Figure 13.6; in this figure the last node, u, on search path for 14 (whose binary representation is 1110) is the node labelled 11?? at level 2 because there is no node labelled 111? at level 3. Thus, we can label each node at level i with an i-bit integer. Then the node u we are searching for is at or 211 13. Data Structures for Integers 13.2. XFastTrie: Searching in Doubly-Logarithmic Time ???? 0 1 0? ? ? 1? ? ? 1 1 00?? 01?? 10?? 11?? 2 1 000? 001? 010? 011? 100? 101? 110? 111? 3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 4 Figure 13.6: The search path for 14 (1110) ends at the node labelled 11?? since there is no node labelled 111?. below level i if and only if there is a node at level i whose label matches the highest-order i bits of x. In an XFastTrie, we store, for each i ∈ {0, . . . , w}, all the nodes at level i in a USet, t[i], that is implemented as a hash table (Chapter 5). Using this USet allows us to check in constant expected time if there is a node at level i whose label matches the highest-order i bits of x. In fact, we can even find this node using t[i].find(x >> (w − i)). The hash tables t[0], . . . , t[w] allow us to use binary search to find u. Initially, we know that u is at some level i with 0 ≤ i < w + 1. We therefore initialize l = 0 and h = w + 1 and repeatedly look at the hash table t[i], where i = b(l + h)/2c. If t[i] contains a node whose label matches x’s highest-order i bits then we set l = i (u is at or below level i), otherwise we set h = i (u is above level i). This process terminates when h − l ≤ 1, in which case we determine that u is at level l. We then complete the find(x) operation using u.jump and the doubly-linked list of leaves. XFastTrie T find(T x) { int l = 0, h = w+1; unsigned ix = intValue(x); Node *v, *u = &r; while (h-l > 1) { int i = (l+h)/2; XPair<Node> p(ix >> (w-i)); if ((v = t[i].find(p).u) == NULL) { h = i; } else { 212 13. Data Structures for Integers 13.3. YFastTrie: A Doubly-Logarithmic Time SSet u = v; l = i; } } if (l == w) return u->x; Node *pred = (((ix >> (w-l-1)) & 1) == 1) ? u->jump : u->jump->prev; return (pred->next == &dummy) ? NULL : pred->next->x; } Each iteration of the while loop in the above method decreases h − l by roughly a factor of 2, so this loop finds u after O(log w) iterations. Each iteration performs a constant amount of work and one find(x) operation in a USet, which takes constant expected time. The remaining work takes only constant time, so the find(x) method in an XFastTrie takes only O(log w) expected time. The add(x) and remove(x) methods for an XFastTrie are almost identical to the same methods in a BinaryTrie. The only modifications are for managing the hash tables t[0],. . . ,t[w]. During the add(x) operation, when a new node is created at level i, this node is added to t[i]. During a remove(x) operation, when a node is removed form level i, this node is removed from t[i]. Since adding and removing from a hash table take constant expected time, this does not increase the running times of add(x) and remove(x) by more than a constant factor. We omit a code listing or add(x) and remove(x) since it is almost identical to the (long) code listing already provided for the same methods in a BinaryTrie. The following theorem summarizes the performance of an XFastTrie: Theorem 13.2. An XFastTrie implements the SSet interface for w-bit integers. An XFastTrie supports the operations • add(x) and remove(x) in O(w) time per operation and • find(x) in O(log w) time per operation. The space used by an XFastTrie that stores n values is O(n · w). 13.3 YFastTrie: A Doubly-Logarithmic Time SSet The XFastTrie is a big improvement over the BinaryTrie in terms of query time—some would even call it an exponential improvement—but the add(x) and remove(x) operations are still not terribly fast. Furthermore, the space usage, O(n · w), is higher than the other SSet implementation in this book, which all use O(n) space. These two problems are related; 213 13. Data Structures for Integers 13.3. YFastTrie: A Doubly-Logarithmic Time SSet ???? 0? ? ? 1? ? ? 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0, 1, 3 4, 5, 8, 9 10, 11, 13 Figure 13.7: A YFastTrie containing the values 0, 1, 3, 4, 6, 8, 9, 10, 11, and 13. if n add(x) operations build a structure of size n · w then the add(x) operation requires on the order of w time (and space) per operation. The YFastTrie data structure simultaneously addresses both the space and speed issues of XFastTries. A YFastTrie uses an XFastTrie, xft, but only stores O(n/w) values in xft. In this way, the total space used by xft is only O(n). Furthermore, only one out of every w add(x) or remove(x) operations in the YFastTrie results in an add(x) or remove(x) operation in xft. By doing this, the average cost incurred by calls to xft’s add(x) and remove(x) operations is only constant. The obvious question becomes: If xft only stores n/w elements, where do the re- maining n(1 − 1/w) elements go? These elements go into secondary structures, in this case an extended version of treaps (Section 7.2). There are roughly n/w of these secondary struc- tures so, on average, each of them stores O(w) items. Treaps support logarithmic time SSet operations, so the operations on these treaps will run in O(log w) time, as required. More concretely, a YFastTrie contains an XFastTrie, xft, that contains a ran- dom sample of the data, where each element appears in the sample independently with probability 1/w. For convenience, the value 2w − 1, is always contained in xft. Let x0 < x1 < · · · < xk−1 denote the elements stored in xft. Associated with each element, xi , is a treap, ti , that stores all values in the range xi−1 + 1, . . . , xi . This is illustrated in Figure 13.7. 214 13. Data Structures for Integers 13.3. YFastTrie: A Doubly-Logarithmic Time SSet The find(x) operation in a YFastTrie is fairly easy. We search for x in xft and find some value xi associated with the treap ti . When then use the treap find(x) method on ti to answer the query. The entire method is a one-liner: YFastTrie T find(T x) { return xft.find(YPair<T>(intValue(x))).t->find(x); } The first find(x) operation (on xft) takes O(log w) time. The second find(x) operation (on a treap) takes O(log r) time, where r is the size of the treap. Later in this section, we will show that the expected size of the treap is O(w) so that this operation takes O(log w) time.1 Adding an element to a YFastTrie is also fairly simple—most of the time. The add(x) method calls xft.find(x) to locate the treap, t, into which x should be inserted. It then calls t.add(x) to add x to t. At this point, it tosses a biased coin, that comes up heads with probability 1/w. If this coin comes up heads, x will be added to xft. This is where things get a little more complicated. When x is added to xft, the treap t needs to be split into two treaps t1 and t0 . The treap t1 contains all the values less than or equal to x; t0 is the original treap, t, with the elements of t1 removed. Once this is done we add the pair (x, t1) to xft. Figure 13.8 shows an example. YFastTrie bool add(T x) { unsigned ix = intValue(x); Treap1<T> *t = xft.find(YPair<T>(ix)).t; if (t->add(x)) { n++; if (rand() % w == 0) { Treap1<T> *t1 = (Treap1<T>*)t->split(x); xft.add(YPair<T>(ix, t1)); } return true; } return false; return true; } Adding x to t takes O(log w) time. Exercise 7.5 shows that splitting t into t1 and t0 can also be done in O(log w) expected time. Adding the pair (x,t1) to xft takes O(w) time, 1 This is an application of Jensen’s Inequality: If E[r] = w, then E[log r] ≤ log w. 215 13. Data Structures for Integers 13.3. YFastTrie: A Doubly-Logarithmic Time SSet ???? 0? ? ? 1? ? ? 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0, 1, 2, 3 4, 5, 6 4, 5,8, 8,99 10, 11, 13 Figure 13.8: Adding the values 2 and 6 to a YFastTrie. The coin toss for 6 came up heads, so 6 was added to xft and the treap containing 4, 5, 6, 8, 9 was split. but only happens with probability 1/w. Therefore, the expected running time of the add(x) operation is 1 O(log w) + O(w) = O(log w) w The remove(x) method just undoes the work performed by add(x). We use xft to find the leaf, u, in xft that contains the answer to xft.find(x). From u, we get the treap, t, containing x and remove x from t. If x was also stored in xft (and x is not equal to 2w − 1) then we remove x from xft and add the elements from x’s treap to the treap, t2, that is stored by u’s successor in the linked list. This is illustrated in Figure 13.9. YFastTrie bool remove(T x) { unsigned ix = intValue(x); XFastTrieNode1<YPair<T> > *u = xft.findNode(ix); bool ret = u->x.t->remove(x); if (ret) n--; if (u->x.ix == ix && ix != UINT_MAX) { Treap1<T> *t2 = u->child[1]->x.t; t2->absorb(*u->x.t); xft.remove(u->x); } return ret; } 216 13. Data Structures for Integers 13.3. YFastTrie: A Doubly-Logarithmic Time SSet ???? 0? ? ? 1? ? ? 00?? 01?? 10?? 11?? 000? 001? 010? 011? 100? 101? 110? 111? 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0,1,2,3 4, 5, 6 8, 9 8, 10, 11, 13 Figure 13.9: Removing the values 1 and 9 from a YFastTrie in Figure 13.8. Finding the node u in xft takes O(log w) expected time. Removing x from t takes O(log w) expected time. Again, Exercise 7.5 shows that merging all the elements of t into t2 can be done in O(log w) time. If necessary, removing x from xft takes O(w) time, but x is only contained in xft with probability 1/w. Therefore, the expected time to remove an element from a YFastTrie is O(log w). Earlier in the discussion, we put off arguing about the sizes of treaps in this structure until later. Before finishing we prove the result we need. Lemma 13.1. Let x be an integer stored in a YFastTrie and let nx denote the number of elements in the treap, t, that contains x. Then E[nx ] ≤ 2w − 1. Proof. Refer to Figure 13.10. Let x1 < x2 < · · · < xi = x < · · · < xn denote the elements stored in the YFastTrie. The treap t contains some elements greater than or equal to x. These are xi , xi+1 , . . . , xi+j−1 , where xi+j−1 is the only one of these elements in which the biased coin toss performed in the add(x) method came up heads. In other words, E[j] is equal to the expected number of biased coin tosses required to obtain the first heads.2 Each coin toss is independent and comes up heads with probability 1/w, so E[j] ≤ w. (See Lemma 4.2 for an analysis of this for the case w = 2.) 2 This analysis ignores the fact that j never exceeds n − i + 1. However, this only decreases E[j], so the upper bound still holds 217 13. Data Structures for Integers 13.4. Discussion and Exercises elements in treap, t, containing x z }| { H T T ... T T T T T ... T H xi−k−1 xi−k xi−k+1 ... xi−2 xi−1 xi = x xi+1 xi+2 ... xi+j−2 xi+j−1 | {z } | {z } k j Figure 13.10: The number of elements in the treap, t, containing x is determined by two coin tossing experiments. Similarly, the elements of t smaller than x are xi−1 , . . . , xi−k where all these k coin tosses come up tails and the coin toss for xi−k−1 comes up heads. Therefore, E[k] ≤ w − 1, since this is the same coin tossing experiment considered previously, but in which the last toss doesn’t count. In summary, nx = j + k, so E[nx ] = E[j + k] = E[j] + E[k] ≤ 2w − 1 . Lemma 13.1 was the last piece in the proof of the following theorem, which summa- rizes the performance of the YFastTrie: Theorem 13.3. A YFastTrie implements the SSet interface for w-bit integers. A YFastTrie supports the operations add(x), remove(x), and find(x) in O(log w) time per operation. The space used by a YFastTrie that stores n values is O(n + w). The w term in the space requirement comes from the fact that xft always stores the value 2w − 1. The implementation could be modified (at the expense of adding some extra cases to the code) so that it is unnecessary to store this value. In this case, the space requirement in the theorem becomes O(n). 13.4 Discussion and Exercises The first data structure to provide O(log w) time add(x), remove(x), and find(x) operations was proposed by van Emde Boas and has since become known as the van Emde Boas (or stratified ) tree. The original van Emde Boas structure had size 2w , so was impractical for large integers. The XFastTrie and YFastTrie data structures were discovered by Willard [62]. The XFastTrie structure is very closely related to van Emde Boas trees. One view of this is that the hash tables in an XFastTrie replace arrays in a van Emde Boas tree. That is, instead of storing the hash table t[i], a van Emde Boas tree stores an array of length 2i . 218 13. Data Structures for Integers 13.4. Discussion and Exercises Another structure for storing integers is Fredman and Willard’s fusion trees [24]. This structure can store n w-bit integers in O(n) space so that the find(x) operation runs in √ O((log n)/(log w)) time. By using a fusion tree when log w > log n and a YFastTrie when √ log w ≤ log n one obtains an O(n) space data structure that can implement the find(x) √ operation in O( log n) time. Recent lower-bound results of Pǎtraşcu and Thorup [48] show that these results are optimal, at least for structures that use only O(n) space. Exercise 13.1. Design and implement a simplified version of a BinaryTrie that doesn’t have a linked list or jump pointers, but can still do find(x) in O(w) time. Exercise 13.2. 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