算法导论：权威指南

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"《算法导论》英文版是计算机科学领域的一部权威著作，由Thomas H. Cormen、Charles E. Leiserson、Ronald L. Rivest和Clifford Stein四位专家共同撰写。该书以其严谨和全面的特性而闻名，被誉为与Donald E. Knuth的《计算机程序设计艺术》齐名的经典之作。" 《算法导论》第三版深入浅出地介绍了算法的基础知识和高级主题，是学习和理解算法的重要参考资料。全书涵盖了广泛的算法问题，包括排序、搜索、图算法、动态规划、数据结构以及计算复杂性理论等核心概念。它不仅提供了算法的详细描述，还解释了如何分析算法的效率，以评估它们在实际应用中的性能。在"ALGORITHMS"这一章节，作者们引出了算法这个核心主题，强调了算法在计算机科学中的核心地位。他们讨论了算法设计的基本原则，如分治法、贪心法、回溯法和动态规划，并通过实例演示了这些方法的运用。此外，书中还包含了算法实现的技巧，以及如何用伪代码或特定编程语言来表述算法。书中深入探讨了诸如快速排序、归并排序、二分查找等经典算法，以及Dijkstra算法、Floyd-Warshall算法等解决图问题的方法。对于复杂度分析，作者们详细解释了时间复杂度和空间复杂度的概念，帮助读者理解算法效率的衡量标准。此外，书中还涉及到了NP完全问题和近似算法，探讨了在面对复杂问题时如何寻找有效解的策略。《算法导论》第三版还包括了关于数据结构的广泛内容，如链表、树、堆、哈希表等，这些都是理解和实现高效算法的关键。同时，书中还介绍了图数据结构和与其相关的操作，这对于网络、物流、路由等问题的解决至关重要。本书适合计算机科学的学生、研究人员和专业开发者阅读，无论你是初学者还是经验丰富的从业者，都能从中受益匪浅。通过深入学习《算法导论》，读者能够提升算法设计和分析能力，从而更好地解决实际问题，提升软件系统和应用的性能。

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

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 f or the third edition

What has changed between the second and third editions of this book? The mag-

nitude of the changes is on a par with the changes between the ﬁrst and second

editions. As we said about the second-edition changes, depending on how you

look at it, the book changed either not much or quite a bit.

A quick look at the table of contents shows that most of the second-edition chap-

ters and sections appear in the third edition. We removed two chapters and one

section, but we have added three new chapters and two new sections apart from

these new chapters.

We kept the hybrid organization from the ﬁrst two editions. Rather than organiz-

ing chapters by only problem domains or according only to techniques, this book

has elements of both. It contains technique-based chapters on divide-and-conquer,

dynamic programming, greedy algorithms, amortized analysis, NP-Completeness,

and approximation algorithms. But it also has entire parts on sorting, on data

structures for dynamic sets, and on algorithms for graph problems. We ﬁnd that

although you need to know how to apply techniques for designing and analyzing al-

gorithms, problems seldom announce to you which techniques are most amenable

to solving them.

Here is a summary of the most signiﬁcant changes for the third edition:



We added new chapters on van Emde Boas trees and multithreaded algorithms,

and we have broken out material on matrix basics into its own appendix chapter.



We revised the chapter on recurrences to more broadly cover the divide-and-

conquer technique, and its ﬁrst two sections apply divide-and-conquer to solve

two problems. The second section of this chapter presents Strassen’s algorithm

for matrix multiplication, which we have moved from the chapter on matrix

operations.



We removed two chapters that were rarely taught: binomial heaps and sorting

networks. One key idea in the sorting networks chapter, the 0-1 principle, ap-

pears in this edition within Problem 8-7 as the 0-1 sorting lemma for compare-

exchange algorithms. The treatment of Fibonacci heaps no longer relies on

binomial heaps as a precursor.

Preface xvii



We revised our treatment of dynamic programming and greedy algorithms. Dy-

namic programming now leads off with a more interesting problem, rod cutting,

than the assembly-line scheduling problem from the second edition. Further-

more, we emphasize memoization a bit more than we did in the second edition,

and we introduce the notion of the subproblem graph as a way to understand

the running time of a dynamic-programming algorithm. In our opening exam-

ple of greedy algorithms, the activity-selection problem, we get to the greedy

algorithm more directly than we did in the second edition.



The way we delete a node from binary search trees (which includes red-black

trees) now guarantees that the node requested for deletion is the node that is

actually deleted. In the ﬁrst two editions, in certain cases, some other node

would be deleted, with its contents moving into the node passed to the deletion

procedure. With our new way to delete nodes, if other components of a program

maintain pointers to nodes in the tree, they will not mistakenly end up with stale

pointers to nodes that have been deleted.



The material on ﬂow networks now bases ﬂows entirely on edges. This ap-

proach is more intuitive than the net ﬂow used in the ﬁrst two editions.



With the material on matrix basics and Strassen’s algorithm moved to other

chapters, the chapter on matrix operations is smaller than in the second edition.



We have modiﬁed our treatment of the Knuth-Morris-Pratt string-matching al-

gorithm.



We corrected several errors. Most of these errors were posted on our Web site

of second-edition errata, but a few were not.



Based on many requests, we changed the syntax (as it were) of our pseudocode.

We now use “ D” to indicate assignment and “

” to test for equality, just as C,

C++, Java, and Python do. Likewise, we have eliminated the keywords do and

then and adopted “//” as our comment-to-end-of-line symbol. We also now use

dot-notation to indicate object attributes. Our pseudocode remains procedural,

rather than object-oriented. In other words, rather than running methods on

objects, we simply call procedures, passing objects as parameters.



We added 100 new exercises and 28 new problems. We also updated many

bibliography entries and added several new ones.



Finally, we went through the entire book and rewrote sentences, paragraphs,

and sections to make the writing clearer and more active.

xviii Preface

Web site

You can use our Web site, http://mitpress.mit.edu/algorithms/, to obtain supple-

mentary information and to communicate with us. The Web site links to a list of

known errors, solutions to selected exercises and problems, and (of course) a list

explaining the corny professor jokes, as well as other content that we might add.

The Web site also tells you how to report errors or make suggestions.

How we produced this book

Like the second edition, the third edition was produced in L

. We used the

Times font with mathematics typeset using the MathTime Pro 2 fonts. We thank

Michael Spivak from Publish or Perish, Inc., Lance Carnes from Personal TeX,

Inc., and Tim Tregubov from Dartmouth College for technical support. As in the

previous two editions, we compiled the index using Windex, a C program that we

wrote, and the bibliography was produced with B

IBT

X. The PDF ﬁles for this

book were created on a MacBook running OS 10.5.

We drew the illustrations for the third edition using MacDraw Pro, with some

of the mathematical expressions in illustrations laid in with the psfrag package

for L

. Unfortunately, MacDraw Pro is legacy software, having not been

marketed for over a decade now. Happily, we still have a couple of Macintoshes

that can run the Classic environment under OS 10.4, and hence they can run Mac-

Draw Pro—mostly. Even under the Classic environment, we ﬁnd MacDraw Pro to

be far easier to use than any other drawing software for the types of illustrations

that accompany computer-science text, and it produces beautiful output.

Who

knows how long our pre-Intel Macs will continue to run, so if anyone from Apple

is listening: Please create an OS X-compatible version of MacDraw Pro!

Acknowledgments for the third edition

We have been working with the MIT Press for over two decades now, and what a

terriﬁc relationship it has been! We thank Ellen Faran, Bob Prior, Ada Brunstein,

and Mary Reilly for their help and support.

We were geographically distributed while producing the third edition, working

in the Dartmouth College Department of Computer Science, the MIT Computer

We investigated several drawing programs that run under Mac OS X, but all had signiﬁcant short-

comings compared with MacDraw Pro. We brieﬂy attempted to produce the illustrations for this

book with a different, well known drawing program. We found that it took at least ﬁve times as long

to produce each illustration as it took with MacDraw Pro, and the resulting illustrations did not look

as good. Hence the decision to revert to MacDraw Pro running on older Macintoshes.

Preface xix

Science and Artiﬁcial Intelligence Laboratory, and the Columbia University De-

partment of Industrial Engineering and Operations Research. We thank our re-

spective universities and colleagues for providing such supportive and stimulating

environments.

Julie Sussman, P.P.A., once again bailed us out as the technical copyeditor. Time

and again, we were amazed at the errors that eluded us, but that Julie caught. She

also helped us improve our presentation in several places. If there is a Hall of Fame

for technical copyeditors, Julie is a sure-ﬁre, ﬁrst-ballot inductee. She is nothing

short of phenomenal. Thank you, thank you, thank you, Julie! Priya Natarajan also

found some errors that we were able to correct before this book went to press. Any

errors that remain (and undoubtedly, some do) are the responsibility of the authors

(and probably were inserted after Julie read the material).

The treatment for van Emde Boas trees derives from Erik Demaine’s notes,

which were in turn inﬂuenced by Michael Bender. We also incorporated ideas

from Javed Aslam, Bradley Kuszmaul, and Hui Zha into this edition.

The chapter on multithreading was based on notes originally written jointly with

Harald Prokop. The material was inﬂuenced by several others working on the Cilk

project at MIT, including Bradley Kuszmaul and Matteo Frigo. The design of the

multithreaded pseudocode took its inspiration from the MIT Cilk extensions to C

and by Cilk Arts’s Cilk++ extensions to C++.

We also thank the many readers of the ﬁrst and second editions who reported

errors or submitted suggestions for how to improve this book. We corrected all the

bona ﬁde errors that were reported, and we incorporated as many suggestions as

we could. We rejoice that the number of such contributors has grown so great that

we must regret that it has become impractical to list them all.

Finally, we thank our wives—Nicole Cormen, Wendy Leiserson, Gail Rivest,

and Rebecca Ivry—and our children—Ricky, Will, Debby, and Katie Leiserson;

Alex and Christopher Rivest; and Molly, Noah, and Benjamin Stein—for their love

and support while we prepared this book. The patience and encouragement of our

families made this project possible. We affectionately dedicate this book to them.

HOMAS H. CORMEN Lebanon, New Hampshire

CHARLES E. LEISERSON Cambridge, Massachusetts

RONALD L. RIVEST Cambridge, Massachusetts

CLIFFORD STEIN New York, New York

February 2009

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