Algorith Analysis Text Book
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 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 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 3.2 Standard notations and common functions
2
3
53
4
?
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 Probabilistic
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 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 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 3.2 Standard notations and common functions
2
3
53
4
?
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 Probabilistic