算法导论：第三版精华概览

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"算法导论-introduction to algorithms 3rd edition" 本书是《算法导论》的第三版，是一本深入介绍计算机算法的经典著作。作者包括Thomas H. Cormen、Charles E. Leiserson、Ronald L. Rivest和Clifford Stein。这本书详细讲解了算法在计算中的作用、算法设计和分析的基础，以及各种重要算法的实现。第一部分（I Foundations）介绍了算法的基本概念和设计方法。第1章讨论了算法的定义和它作为技术的角色，第2章通过插入排序来引入算法分析和设计，第3章讲解了渐进表示法，如大O符号，以及常见的函数增长，第4章则引入了分治策略，通过最大子数组问题、矩阵乘法的Strassen算法以及递归方程的解决方法（如回溯树方法和主定理）进行讲解。第5章涵盖了概率分析和随机算法，如雇佣问题、指示随机变量以及随机化算法的分析。第二部分（II Sorting and Order Statistics）探讨了排序和顺序统计问题。第6章介绍了堆排序和堆数据结构，第7章讲解快速排序及其性能，第8章讨论线性时间内的排序算法，如计数排序、基数排序和桶排序，第9章涉及中位数和顺序统计，包括最小值和最大值的选择。第三部分（III Data Structures）详细讲述了数据结构。第10章涉及基本数据结构，如栈、队列和链表，第11章讲解哈希表，包括直接寻址表和开放寻址，第12章介绍了二叉搜索树，第13章是关于红黑树，第14章讨论了增强数据结构，如动态顺序统计和区间树。第四部分（IV Advanced Design and Analysis Techniques）深入了高级设计和分析技术。第15章介绍了动态规划，如剪刀剪纸和矩阵链乘法，第16章讲解了贪心算法，包括活动选择问题和霍夫曼编码，第17章介绍了摊还分析，如动态表格。第五部分（V Advanced Data Structures）涵盖了更复杂的数据结构。第18章涉及B树，第19章是关于斐波那契堆，第20章讲解了van Emde Boas树，第21章介绍了用于不相交集合的数据结构。第六部分（VI Graph Algorithms）探讨图算法。第22章介绍了图的表示和基本算法，如广度优先搜索和深度优先搜索，第23章讲解了最小生成树，第24章讨论单源最短路径问题。《算法导论》第三版是一本全面的教材，适合计算机科学和工程专业的学生，以及任何对算法有深入研究需求的读者。书中通过实例和详细的解释，使读者能够理解和掌握各种重要的算法和数据结构，同时提供了理论分析和实际应用的平衡。

Preface xv

What are the prerequisites for reading this book?

You should h ave some programming experie nce. In partic ular, y ou should u n-

derstand recursive procedures and simple data structures such as arrays and

linked lists.

You should have some facility w ith mat hematical proofs , and espe cially 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

afewoftheproblemsandexercises. Feelfreetocheckyoursolutionsagainstours.

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 for 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.

Aquicklookatthetableofcontentsshowsthatmostofthesecond-editionchap-

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”toindicateassignmentand“

”totestforequality,justasC,

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

then and adopted “//”asourcomment-to-end-of-linesymbol.Wealsonowuse

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 c an use our Web site, http://mitpres s.m it.edu/algorithms/, to obtain s u pple-

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 p roduced this book

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

.Weusedthe

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,MacDrawProislegacysoftware,havingnotbeen

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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