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T H O M A S H. C O R M E N C H A R L E S E. L E I S E R S O N R O N A L D L. R I V E S T C L I F F O R D STEIN

INTRODUCTION TO

ALGORITHMS
T H I R D E D I T I O N

Introduction to Algorithms
Third Edition

Thomas H. Cormen Charles E. Leiserson Ronald L. Rivest Clifford Stein

Introduction to Algorithms
Third Edition

The MIT Press Cambridge, Massachusetts

London, England c 2009 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 inTimes 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. Computeralgorithms. I. Cormen, Thomas H. QA76.6.I5858 2009 005.1—dc22 2009008593 10 9 8 7 6 5 4 3 2

Contents

Preface

xiii

I Foundations
Introduction 1 3 The Role of Algorithms in Computing 5 1.1 Algorithms 5 1.2 Algorithms as a technology 11 Getting Started 16 2.1 Insertion sort 16 2.2 Analyzing algorithms 23 2.3 Designing algorithms 29 Growth of Functions 43 3.1 Asymptotic notation 43 3.2Standard notations and common functions

2

3

53

4

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5

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 ProbabilisticAnalysis 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

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vi

Contents

II Sorting and Order Statistics
Introduction 6 147

Heapsort 151 6.1 Heaps 151 6.2 Maintaining the heap property 6.3 Building a heap 156 6.4 The heapsort algorithm 1596.5 Priority queues 162

154

7

Quicksort 170 7.1 Description of quicksort 170 7.2 Performance of quicksort 174 7.3 A randomized version of quicksort 7.4 Analysis of quicksort 180 Sorting in Linear Time 191 8.1 Lower bounds for sorting 8.2 Counting sort 194 8.3 Radix sort 197 8.4 Bucket sort 200 191

179

8

9

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

III Data Structures
Introduction 10 229 Elementary Data Structures 232 10.1 Stacks and queues 232 10.2 Linked lists 236 10.3 Implementing pointers and objects 10.4 Representing rooted trees 246 Hash Tables 253 11.1 Direct-address tables 254 11.2 Hash tables 256 11.3 Hash functions 262 11.4 Open addressing 269 11.5Perfect hashing 277

241

11

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Contents

vii

12

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13

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

14

Augmenting Data Structures339 14.1 Dynamic order statistics 339 14.2 How to augment a data structure 14.3 Interval trees 348

345

IV Advanced Design and Analysis Techniques
Introduction 15 357

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 Greedy Algorithms 414 16.1...
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