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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 S T E I N
I N T R O D U C T I O N T O
A L G O R I T H M S
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 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
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
Per ............

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