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Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher.

For information about special quantity discounts, please email special sales@mitpress.mit.edu.

This book was set in Times Roman and Mathtime Pro 2 by the authors.

Printed and bound in the United States of America.

Library of Congress Cataloging-in-Publication Data

Introduction to algorithms / Thomas H. Cormen . . . [et al.].—3rd ed. p. cm.

Includes bibliographical references and index. ISBN 978-0-262-03384-8 (hardcover : alk. paper)—ISBN 978-0-262-53305-8 (pbk. : alk. paper) 1. Computer programming. 2. Computer algorithms. I. Cormen, Thomas H.

QA76.6.I5858 2009 005.1—dc22

2009008593

10 9 8 7 6 5 4 3 2

mailto:sales@mitpress.mit.edu

Contents

Preface xiii

I Foundations

Introduction 3

1 The Role of Algorithms in Computing 5 1.1 Algorithms 5 1.2 Algorithms as a technology 11

2 Getting Started 16 2.1 Insertion sort 16 2.2 Analyzing algorithms 23 2.3 Designing algorithms 29

3 Growth of Functions 43 3.1 Asymptotic notation 43 3.2 Standard notations and common functions 53

4 Divide-and-Conquer 65 4.1 The maximum-subarray problem 68 4.2 Strassen’s algorithm for matrix multiplication 75 4.3 The substitution method for solving recurrences 83 4.4 The recursion-tree method for solving recurrences 88 4.5 The master method for solving recurrences 93

? 4.6 Proof of the master theorem 97

5 Probabilistic Analysis and Randomized Algorithms 114 5.1 The hiring problem 114 5.2 Indicator random variables 118 5.3 Randomized algorithms 122

? 5.4 Probabilistic analysis and further uses of indicator random variables 130

vi Contents

II Sorting and Order Statistics

Introduction 147

6 Heapsort 151 6.1 Heaps 151 6.2 Maintaining the heap property 154 6.3 Building a heap 156 6.4 The heapsort algorithm 159 6.5 Priority queues 162

7 Quicksort 170 7.1 Description of quicksort 170 7.2 Performance of quicksort 174 7.3 A randomized version of quicksort 179 7.4 Analysis of quicksort 180

8 Sorting in Linear Time 191 8.1 Lower bounds for sorting 191 8.2 Counting sort 194 8.3 Radix sort 197 8.4 Bucket sort 200

9 Medians and Order Statistics 213 9.1 Minimum and maximum 214 9.2 Selection in expected linear time 215 9.3 Selection in worst-case linear time 220

III Data Structures

Introduction 229

10 Elementary Data Structures 232 10.1 Stacks and queues 232 10.2 Linked lists 236 10.3 Implementing pointers and objects 241 10.4 Representing rooted trees 246

11 Hash Tables 253 11.1 Direct-address tables 254 11.2 Hash tables 256 11.3 Hash functions 262 11.4 Open addressing 269

? 11.5 Perfect hashing 277

Contents vii

12 Binary Search Trees 286 12.1 What is a binary search tree? 286 12.2 Querying a binary search tree 289 12.3 Insertion and deletion 294

? 12.4 Randomly built binary search trees 299

13 Red-Black Trees 308 13.1 Properties of red-black trees 308 13.2 Rotations 312 13.3 Insertion 315 13.4 Deletion 323

14 Augmenting Data Structures 339 14.1 Dynamic order statistics 339 14.2 How to augment a data structure 345 14.3 Interval trees 348

IV Advanced Design and Analysis Techniques

Introduction 357

15 Dynamic Programming 359 15.1 Rod cutting 360 15.2 Matrix-chain multiplication 370 15.3 Elements of dynamic programming 378 15.4 Longest common subsequence 390 15.5 Optimal binary search trees 397

16 Greedy Algorithms 414 16.1 An activity-selection problem 415 16.2 Elements of the greedy strategy 423 16.3 Huffman codes 428

? 16.4 Matroids and greedy methods 437 ? 16.5 A task-scheduling problem as a matroid 443

17 Amortized Analysis 451 17.1 Aggregate analysis 452 17.2 The accounting method 456 17.3 The potential method 459 17.4 Dynamic tables 463

viii Contents

V Advanced Data Structures

Introduction 481

18 B-Trees 484 18.1 Definition of B-trees 488 18.2 Basic operations on B-trees 491 18.3 Deleting a key from a B-tree 499

19 Fibonacci Heaps 505 19.1 Structure of Fibonacci heaps 507 19.2 Mergeable-heap operations 510 19.3 Decreasing a key and deleting a node 518 19.4 Bounding the maximum degree 523

20 van Emde Boas Trees 531 20.1 Preliminary approaches 532 20.2 A recursive structure 536 20.3 The van Emde Boas tree 545

21 Data Structures for Disjoint Sets 561 21.1 Disjoint-set operations 561 21.2 Linked-list representation of disjoint sets 564 21.3 Disjoint-set forests 568

? 21.4 Analysis of union by rank with path compression 573

VI Graph Algorithms

Introduction 587

22 Elementary Graph Algorithms 589 22.1 Representations of graphs 589 22.2 Breadth-first search 594 22.3 Depth-first search 603 22.4 Topological sort 612 22.5 Strongly connected components 615

23 Minimum Spanning Trees 624 23.1 Growing a minimum spanning tree 625 23.2 The algorithms of Kruskal and Prim 631

Contents ix

24 Single-Source Shortest Paths 643 24.1 The Bellman-Ford algorithm 651 24.2 Single-source shortest paths in directed acyclic graphs 655 24.3 Dijkstra’s algorithm 658 24.4 Difference constraints and shortest paths 664 24.5 Proofs of shortest-paths properties 671

25 All-Pairs Shortest Paths 684 25.1 Shortest paths and matrix multiplication 686 25.2 The Floyd-Warshall algorithm 693 25.3 Johnson’s algorithm for sparse graphs 700

26 Maximum Flow 708 26.1 Flow networks 709 26.2 The Ford-Fulkerson method 714 26.3 Maximum bipartite matching 732

? 26.4 Push-relabel algorithms 736 ? 26.5 The relabel-to-front algorithm 748

VII Selected Topics

Introduction 769

27 Multithreaded Algorithms 772 27.1 The basics of dynamic multithreading 774 27.2 Multithreaded matrix multiplication 792 27.3 Multithreaded merge sort 797

28 Matrix Operations 813 28.1 Solving systems of linear equations 813 28.2 Inverting matrices 827 28.3 Symmetric positive-definite matrices and least-squares approximation

832

29 Linear Programming 843 29.1 Standard and slack forms 850 29.2 Formulating problems as linear programs 859 29.3 The simplex algorithm 864 29.4 Duality 879 29.5 The initial basic feasible solution 886

x Contents

30 Polynomials and the FFT 898 30.1 Representing polynomials 900 30.2 The DFT and FFT 906 30.3 Efficient FFT implementations 915

31 Number-Theoretic Algorithms 926 31.1 Elementary number-theoretic notions 927 31.2 Greatest common divisor 933 31.3 Modular arithmetic 939 31.4 Solving modular linear equations 946 31.5 The Chinese remainder theorem 950 31.6 Powers of an element 954 31.7 The RSA public-key cryptosystem 958

? 31.8 Primality testing 965 ? 31.9 Integer factorization 975

32 String Matching 985 32.1 The naive string-matching algorithm 988 32.2 The Rabin-Karp algorithm 990 32.3 String matching with finite automata 995

? 32.4 The Knuth-Morris-Pratt algorithm 1002

33 Computational Geometry 1014 33.1 Line-segment properties 1015 33.2 Determining whether any pair of segments intersects 1021 33.3 Finding the convex hull 1029 33.4 Finding the closest pair of points 1039

34 NP-Completeness 1048 34.1 Polynomial time 1053 34.2 Polynomial-time verification 1061 34.3 NP-completeness and reducibility 1067 34.4 NP-completeness proofs 1078 34.5 NP-complete problems 1086

35 Approximation Algorithms 1106 35.1 The vertex-cover problem 1108 35.2 The traveling-salesman problem 1111 35.3 The set-covering problem 1117 35.4 Randomization and linear programming 1123 35.5 The subset-sum problem 1128

Contents xi

VIII Appendix: Mathematical Background

Introduction 1143

A Summations 1145 A.1 Summation formulas and properties 1145 A.2 Bounding summations 1149

B Sets, Etc. 1158 B.1 Sets 1158 B.2 Relations 1163 B.3 Functions 1166 B.4 Graphs 1168 B.5 Trees 1173

C Counting and Probability 1183 C.1 Counting 1183 C.2 Probability 1189 C.3 Discrete random variables 1196 C.4 The geometric and binomial distributions 1201

? C.5 The tails of the binomial distribution 1208

D Matrices 1217 D.1 Matrices and matrix operations 1217 D.2 Basic matrix properties 1222

Bibliography 1231

Index 1251

Preface

Before there were computers, there were algorithms. But now that there are com- puters, there are even more algorithms, and algorithms lie at the heart of computing.

This book provides a comprehensive introduction to the modern study of com- puter algorithms. It presents many algorithms and covers them in considerable depth, yet makes their design and analysis accessible to all levels of readers. We have tried to keep explanations elementary without sacrificing depth of coverage or mathematical rigor.

Each chapter presents an algorithm, a design technique, an application area, or a related topic. Algorithms are described in English and in a pseudocode designed to be readable by anyone who has done a little programming. The book contains 244 figures—many with multiple parts—illustrating how the algorithms work. Since we emphasize efficiency as a design criterion, we include careful analyses of the running times of all our algorithms.

The text is intended primarily for use in undergraduate or graduate courses in algorithms or data structures. Because it discusses engineering issues in algorithm design, as well as mathematical aspects, it is equally well suited for self-study by technical professionals.

In this, the third edition, we have once again updated the entire book. The changes cover a broad spectrum, including new chapters, revised pseudocode, and a more active writing style.

To the teacher

We have designed this book to be both versatile and complete. You should find it useful for a variety of courses, from an undergraduate course in data structures up through a graduate course in algorithms. Because we have provided considerably more material than can fit in a typical one-term course, you can consider this book to be a “buffet” or “smorgasbord” from which you can pick and choose the material that best supports the course you wish to teach.

xiv Preface

You should find it easy to organize your course around just the chapters you need. We have made chapters relatively self-contained, so that you need not worry about an unexpected and unnecessary dependence of one chapter on another. Each chapter presents the easier material first and the more difficult material later, with section boundaries marking natural stopping points. In an undergraduate course, you might use only the earlier sections from a chapter; in a graduate course, you might cover the entire chapter.

We have included 957 exercises and 158 problems. Each section ends with exer- cises, and each chapter ends with problems. The exercises are generally short ques- tions that test basic mastery of the material. Some are simple self-check thought exercises, whereas others are more substantial and are suitable as assigned home- work. The problems are more elaborate case studies that often introduce new ma- terial; they often consist of several questions that lead the student through the steps required to arrive at a solution.

Departing from our practice in previous editions of this book, we have made publicly available solutions to some, but by no means all, of the problems and ex- ercises. Our Web site, http://mitpress.mit.edu/algorithms/, links to these solutions. You will want to check this site to make sure that it does not contain the solution to an exercise or problem that you plan to assign. We expect the set of solutions that we post to grow slowly over time, so you will need to check it each time you teach the course.

We have starred (?) the sections and exercises that are more suitable for graduate students than for undergraduates. A starred section is not necessarily more diffi- cult than an unstarred one, but it may require an understanding of more advanced mathematics. Likewise, starred exercises may require an advanced background or more than average creativity.

To the student

We hope that this textbook provides you with an enjoyable introduction to the field of algorithms. We have attempted to make every algorithm accessible and interesting. To help you when you encounter unfamiliar or difficult algorithms, we describe each one in a step-by-step manner. We also provide careful explanations of the mathematics needed to understand the analysis of the algorithms. If you already have some familiarity with a topic, you will find the chapters organized so that you can skim introductory sections and proceed quickly to the more advanced material.

This is a large book, and your class will probably cover only a portion of its material. We have tried, however, to make this a book that will be useful to you now as a course textbook and also later in your career as a mathematical desk reference or an engineering handbook.

http://mitpress.mit.edu/algorithms/

Preface xv

What are the prerequisites for reading this book?

� You should have some programming experience. In particular, you should un- derstand recursive procedures and simple data structures such as arrays and linked lists.

� You should have some facility with mathematical proofs, and especially proofs by mathematical induction. A few portions of the book rely on some knowledge of elementary calculus. Beyond that, Parts I and VIII of this book teach you all the mathematical techniques you will need.

We have heard, loud and clear, the call to supply solutions to problems and exercises. Our Web site, http://mitpress.mit.edu/algorithms/, links to solutions for a few of the problems and exercises. Feel free to check your solutions against ours. We ask, however, that you do not send your solutions to us.

To the professional

The wide range of topics in this book makes it an excellent handbook on algo- rithms. Because each chapter is relatively self-contained, you can focus in on the topics that most interest you.

Most of the algorithms we discuss have great practical utility. We therefore address implementation concerns and other engineering issues. We often provide practical alternatives to the few algorithms that are primarily of theoretical interest.

If you wish to implement any of the algorithms, you should find the transla- tion of our pseudocode into your favorite programming language to be a fairly straightforward task. We have designed the pseudocode to present each algorithm clearly and succinctly. Consequently, we do not address error-handling and other software-engineering issues that require specific assumptions about your program- ming environment. We attempt to present each algorithm simply and directly with- out allowing the idiosyncrasies of a particular programming language to obscure its essence.

We understand that if you are using this book outside of a course, then you might be unable to check your solutions to problems and exercises against solutions provided by an instructor. Our Web site, http://mitpress.mit.edu/algorithms/, links to solutions for some of the problems and exercises so that you can check your work. Please do not send your solutions to us.

To our colleagues

We have supplied an extensive bibliography and pointers to the current literature. Each chapter ends with a set of chapter notes that give historical details and ref- erences. The chapter notes do not provide a complete reference to the whole field

http://mitpress.mit.edu/algorithms/

http://mitpress.mit.edu/algorithms/

xvi Preface

of algorithms, however. Though it may be hard to believe for a book of this size, space constraints prevented us from including many interesting algorithms.

Despite myriad requests from students for solutions to problems and exercises, we have chosen as a matter of policy not to supply references for problems and exercises, to remove the temptation for students to look up a solution rather than to find it themselves.

Changes for the third edition

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