算法导论：探索计算机科学的核心步骤

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"算法导论(第三版)" 是一本关于算法理论和实践的权威教材，由Thomas H. Cormen、Charles E. Leiserson、Ronald L. Rivest和Clifford Stein四位专家合著。本书深入浅出地介绍了算法的设计、分析以及其在计算机科学中的应用。在计算机科学中，算法是至关重要的概念。简单来说，算法就是一个明确的计算过程，它接受一个或一组输入值，并产生一个或一组输出值。这个过程由一系列计算步骤组成，将输入转化为输出。算法也可以视为解决特定计算问题的方法。问题的定义概括了期望的输入/输出关系，而算法则具体描述了实现这一关系的计算步骤。以排序问题为例，这是实际应用中常见的问题，也是引入许多标准设计技术和分析工具的良好起点。排序问题要求我们将一系列数字按非递减顺序排列。解决这个问题需要设计一个算法，该算法能够接受任意顺序的数字序列，然后通过一系列操作将其转换成有序序列。《算法导论》第三版详细探讨了算法的基础和高级主题，包括但不限于排序算法（如快速排序、归并排序）、搜索算法（如二分查找）、图算法（如最短路径算法、最小生成树算法）以及动态规划等经典问题。此外，书中还涵盖了复杂度分析，帮助读者理解算法的时间和空间效率，以及如何评估和比较不同算法的性能。书中还包括了算法设计策略，如分治法、贪心法、回溯法和随机化算法。此外，还讨论了数据结构（如堆、栈、队列、树和图）与算法之间的关系，以及它们如何影响算法的实现和效率。这些内容不仅适用于理论学习，也对实际编程和软件开发具有指导意义。通过大量的例子、练习和实例，本书旨在培养读者理解和解决问题的能力，提高他们在实际工程中应用算法的技能。此外，书中的习题涵盖了从基础到高级的各种难度，适合不同水平的读者进行自我测试和提升。《算法导论(第三版)》是计算机科学领域的一本经典教材，无论对于初学者还是经验丰富的专业人士，都能从中受益匪浅，深入理解和掌握算法的核心原理及其应用。

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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