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The bible of all fundamental algorithms and the work that taught many of today's software developers most of what they know about computer programming.
–Byte, September 1995
I can't begin to tell you how many pleasurable hours of study and recreation they have afforded me! I have pored over them in cars, restaurants, at work, at home... and even at a Little League game when my son wasn't in the line-up.
–Charles Long
If you think you're a really good programmer... read [Knuth's] Art of Computer Programming... You should definitely send me a resume if you can read the whole thing.
–Bill Gates
It's always a pleasure when a problem is hard enough that you have to get the Knuths off the shelf. I find that merely opening one has a very useful terrorizing effect on computers.
–Jonathan Laventhol
This first volume in the series begins with basic programming concepts and techniques, then focuses more particularly on information structures–the representation of information inside a computer, the structural relationships between data elements and how to deal with them efficiently. Elementary applications are given to simulation, numerical methods, symbolic computing, software and system design. Dozens of simple and important algorithms and techniques have been added to those of the previous edition. The section on mathematical preliminaries has been extensively revised to match present trends in research.
Ebook (PDF version) produced by Mathematical Sciences Publishers (MSP),http://msp.org
Innovations: Where are those developments reflected in these new editions?
Knuth: I've gone over every page and updated material, when I think the
subject has "converged" to a form that people will find important not only today
but also 50 or 100 years from now. Such changes appear throughout the books,
most notably in the chapter on random numbers. On the other hand, many topics
in Volumes 1, 2, and 3 are still evolving rapidly. In such cases, I have not
made a major update; I've simply added a little icon to the page, meaning "sorry,
still under construction"! I will do a final update to those books after I finish
Volumes 4 and 5; otherwise I'd have to rewrite them again, and I would never
finish. It's more important for me to get Volume 4 done than to keep Volumes
1, 2, and 3 strictly up to the minute.
The new editions have hundreds of new exercises and answers to exercises
that I know will always be instructive; I've been noting these things in my
own copies of the books since the 1970s, and I'm making them public now.
Innovations: Why revise Volumes 1, 2, & 3 before publishing Volume
4?
Knuth:Because they haven't been revised for a long time and I have a
megabyte of updates that I'm sure people will want to know about. Silvio Levy
has made it possible for me to do this without taking much time away from Volume
4, because he's doing the hard work of converting the old books to TeX and merging
everything together. Another friend, Jeff Oldham, is putting all the illustrations
into METAPOST form, so that they will be improved too.
And there's another significant reason: By going through Volumes
1, 2, and 3 in this way, I'm able to be sure that Volume 4 matches them well,
in spite of the fact that I took 13 years off to work on TeX and METAFONT and
Concrete Mathematics and some other books that had to be written in the 80s.
"I've gone over every page and updated material, when I think the subject has 'converged' to a form that people will find important not only today but also 50 or 100 years from now."Innovations: Do you still see this as a seven volume set?
Innovations: Can you tell us about the process by which Volume 4 will
eventually be published?
Knuth: I'll publish so-called fascicles, about 128 pages each, about
twice a year. These will be "beta-test" versions of the eventual book; they
will represent my best shot, but I'm sure that readers will be able to help
me make many improvements in the final edition. The subject is so vast that
I cannot hope to get everything right on my first try. Charles Dickens did a
similar thing with his novels: He published fascicles containing Chapters 1
and 2 before he had any idea how the stories were going to end. That way he
could get the best reader feedback.
I view my role as trying to be a spokesman for many people who are
developing computer science; I try to present their discoveries in a uniform
way that a programmer-on-the-street who cannot read advanced scientific jargon
will be able to understand. I've spent 35 years gathering a database of materials
and notes about these topics, and I think my point of view (although biased)
will be helpful to many readers; that's why I'm hoping to have readers participate
and have adopted a fascicle-preview strategy.
Innovations: What inspired you to start this project?
Knuth:There was no reliable guide to the literature in 1962. I was the
only person I knew who had read most of the journals yet had not discovered
very many things myself; and I liked to write. Thus I thought I could give a
more balanced and unbiased account than the people who had made the most important
discoveries. Of course, after I got going, I discovered a few things of my own,
so by now I'm as biased as anybody. But you asked about what was the inspiration
in 1962. And the answer is: There was a huge need for a book like The Art of
Computer Programming, but everybody who was capable of writing it was unfortunately
likely to give a terribly slanted account!
Innovations: What do you see as the biggest challenge facing programmers
today?
Knuth: The hardest thing is to go to sleep at night, when there are so
many urgent things needing to be done. A huge gap exists between what we know
is possible with today's machines and what we have so far been able to finish.
Innovations: Who have been the biggest influences on your computing
career?
Knuth: Of course I have been tremendously influenced b
The Art of Computer Programming: Introduction to Algorithms
1. Basic Concepts.
Algorithms.
Mathematical Preliminaries.
Mathematical Induction.
Numbers, Powers, and Logarithms.
Sums and Products.
Integer Functions and Elementary Number Theory.
Permutations and Factorials.
Binomial Coefficients.
Harmonic Numbers.
Fibonacci Numbers.
Generating Functions.
Analysis of an Algorithm.
Asymptotic Representations.
MIX.
Description of MIX.
The MIX Assembly Language.
Applications to Permutations.
Some Fundamental Programming Techniques.
Subroutines.
Coroutines.
Interpretive Routines.
Input and Output.
History and Bibliography.
Introduction.
Linear Lists.
Stacks, Queues, and Deques.
Sequential Allocation.
Linked Allocation.
Circular Lists.
Doubly Linked Lists.
Arrays and Orthogonal Lists.
Trees.
Traversing Binary Trees.
Binary Tree Representation of Trees.
Other Representations of Trees.
Basic Mathematical Properties of Trees.
Lists and Garbage Collection.
Multilinked Structures.
Dynamic Storage Allocation.
History and Bibliography.
1. Fundamental Constants (decimal).
2. Fundamental Constants (octal).
3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers.
Here is your book, the one your thousands of letters have asked us to publish. It has taken us years to do, checking and rechecking countless recipes to bring you only the best, only the interesting, only the perfect. Now we can say, without a shadow of a doubt, that every single one of them, if you follow the directions to the letter, will work for you exactly as well as it did for us, even if you have never cooked before.
—McCall's Cookbook (1963)
The process of preparing programs for a digital computer is especially attractive, not only because it can be economically and scientifically rewarding, but also because it can be an aesthetic experience much like composing poetry or music. This book is the first volume of a multi-volume set of books that has been designed to train the reader in various skillsthat go into a programmer's craft.
The following chapters are not meant to serve as an introduction to computer programming; the reader is supposed to have had some previous experience. The prerequisites are actually very simple, but a beginner requires time and practice in order to understand the concept of a digital computer. The reader should possess:
These four prerequisites can perhaps be summed up into the single requirement that the reader should have already written and tested at least, say, four programs for at least one computer.
I have tried to write this set of books in such a way that it will fill several needs. In the first place, these books are reference works that summarize the knowledge that has been acquired in several important fields. In the second place, they can be used as textbooks for self-study or for college courses in the computer and information sciences. To meet both of these objectives, I have incorporated a large number of exercises into the text and have furnished answers for most of them. I have also made an effort to fill the pages with facts rather than with vague, general commentary.
This set of books is intended for people who will be more than just casually interested in computers, yet it is by no means only for the computer specialist. Indeed, one of my main goals has been to make these programming techniques more accessible to the many people working in other fields who can make fruitful use of computers, yet who cannot afford the time to locate all of the necessary information that is buried in technical journals.
We might call the subject of these books "nonnumerical analysis." Computers have traditionally been associated with the solution of numerical problems such as the calculation of the roots of an equation, numerical interpolation and integration, etc., but such topics are not treated here except in passing. Numerical computer programming is an extremely interesting and rapidly expanding field, and many books have been written about it. Since the early 1960s, however, computers have been used even more often for problems in which numbers occur only by coincidence; the computer's decision-making capabilities are being used, rather than its ability to do arithmetic. We have some use for addition and subtraction in nonnumerical problems, but we rarely feel any need for multiplication and division. Of course, even a person who is primarily concerned with numerical computer programming will benefit from a study of the nonnumerical techniques, for they are present in the background of numerical programs as well.
The results of research in nonnumerical analysis are scattered throughout numerous technical journals. My approach has been to try to distill this vast literature by studying the techniques that are most basic, in the sense that they can be applied to many types of programming situations. I have attempted to coordinate the ideas into more or less of a "theory," as well as to show how the theory applies to a wide variety of practical problems.
Of course, "nonnumerical analysis" is a terribly negative name for this field of study; it is much better to have a positive, descriptive term that characterizes the subject. "Information processing" is too broad a designation for the material I am considering, and "programming techniques" is too narrow. Therefore I wish to propose analysis of algorithms as an appropriate name for the subject matter covered in these books. This name is meant to imply "the theory of the properties of particular computer algorithms."
The complete set of books, entitled The Art of Computer Programming, has the following general outline:
Volume 4 deals with such a large topic, it actually represents three separate books (Volumes 4A, 4B, and 4C). Two additional volumes on more specialized topics are also planned: Volume 6, The Theory of Languages (Chapter 11); Volume 7, Compilers (Chapter 12).
I started out in 1962 to write a single book with this sequence of chapters, but I soon found that it was more important to treat the subjects in depth rather than to skim over them lightly. The resulting length of the text has meant that each chapter by itself contains more than enough material for a one-semester college course; so it has become sensible to publish the series in separate volumes. I know that it is strange to have only one or two chapters in an entire book, but I have decided to retain the original chapter numbering in order to facilitate cross-references. A shorter version of Volumes 1 through 5 is planned, intended specifically to serve as a more general reference and/or text for undergraduate computer courses; its contents will be a subset of the material in these books, with the more specialized information omitted. The same chapter numbering will be used in the abridged edition as in the complete work.
The present volume may be considered as the "intersection" of the entire set, in the sense that it contains basic material that is used in all the other books. Volumes 2 through 5, on the other hand, may be read independently of each other. Volume 1 is not only a reference book to be used in connection with the remaining volumes; it may also be used in college courses or for self-study as a text on the subject of data structures (emphasizing the material of Chapter 2), or as a text on the subject of discrete mathematics (emphasizing the material of Sections 1.1, 1.2, 1.3.3, and 2.3.4), or as a text on the subject of machine-language programming (emphasizing the material of Sections 1.3 and 1.4).
The point of view I have adopted while writing these chapters differs from that taken in most contemporary books about computer programming in that I am not trying to teach the reader how to use somebody else's software. I am concerned rather with teaching people how to write better software themselves.
My original goal was to bring readers to the frontiers of knowledge in every subject that was treated. But it is extremely difficult to keep up with a field that is economically profitable, and the rapid rise of computer science has made such a dream impossible. The subject has become a vast tapestry with tens of thousands of subtle results contributed by tens of thousands of talented people all over the world. Therefore my new goal has been to concentrate on "classic" techniques that are likely to remain important for many more decades, and to describe them as well as I can. In particular, I have tried to trace the history of each subject, and to provide a solid foundation for future progress. I have attempted to choose terminology that is concise and consistent with current usage. I have tried to include all of the known ideas about sequential computer programming that are both beautiful and easy to state.
A few words are in order about the mathematical content of this set of books. The material has been organized so that persons with no more than a knowledge of high-school algebra may read it, skimming briefly over the more mathematical portions; yet a reader who is mathematically inclined will learn about many interesting mathematical techniques related to discrete mathematics. This dual level of presentation has been achieved in part by assigning ratings to each of the exercises so that the primarily mathematical ones are marked specifically as such, and also by arranging most sections so that the main mathematical results are stated before their proofs. The proofs are either left as exercises (with answers to be found in a separate section) or they are given at the end of a section.
A reader who is interested primarily in programming rather than in the associated mathematics may stop reading most sections as soon as the mathematics becomes recognizably difficult. On the other hand, a mathematically oriented reader will find a wealth of interesting material collected here. Much of the published mathematics about computer programming has been faulty, and one of the purposes of this book is to instruct readers in proper mathematical approaches to this subject. Since I profess to be a mathematician, it is my duty to maintain mathematical integrity as well as I can.
A knowledge of elementary calculus will suffice for most of the mathematics in these books, since most of the other theory that is needed is developed herein. However, I do need to use deeper theorems of complex variable theory, probability theory, number theory, etc., at times, and in such cases I refer to appropriate textbooks where those subjects are developed.
The hardest decision that I had to make while preparing these books concerned the manner in which to present the various techniques. The advantages of flow charts and of an informal step-by-step description of an algorithm are well known; for a discussion of this, see the article "Computer-Drawn Flowcharts" in the ACM Communications, Vol. 6 (September 1963), pages 555–563. Yet a formal, precise language is also necessary to specify any computer algorithm, and I needed to decide whether to use an algebraic language, such as ALGOL or FORTRAN, or to use a machine-oriented language for this purpose. Perhaps many of today's computer experts will disagree with my decision to use a machine-oriented language, but I have become convinced that it was definitely the correct choice, for the following reasons:
(b) The programs we require are, with a few exceptions, all rather short, so with a suitable computer there will be no trouble understanding the programs.
(c) High-level languages are inadequate for discussing important low-level details such as coroutine linkage, random number generation, multi-precision arithmetic, and many problems involving the efficient usage of memory.
(d) A person who is more than casually interested in computers should be well schooled in machine language, since it is a fundamental part of a computer.
(e) Some machine language would be necessary anyway as output of the software programs described in many of the examples.
(f) New algebraic languages go in and out of fashion every five years or so, while I am trying to emphasize concepts that are timeless.
From the other point of view, I admit that it is somewhat easier to write programs in higher-level programming languages, and it is considerably easier to debug the programs. Indeed, I have rarely used low-level machine language for my own programs since 1970, now that computers are so large and so fast. Many of the problems of interest to us in this book, however, are those for which the programmer's art is most important. For example, some combinatorial calculations need to be repeated a trillion times, and we save about 11.6 days of computation for every microsecond we can squeeze out of their inner loop. Similarly, it is worthwhile to put an additional effort into the writing of software that will be used many times each day in many computer installations, since the software needs to be written only once.
Given the decision to use a machine-oriented language, which language should be used? I could have chosen the language of a particular machine X, but then those people who do not possess machine X would think this book is only for X-people. Furthermore, machine X probably has a lot of idiosyncrasies that are completely irrelevant to the material in this book yet which must be explained; and in two years the manufacturer of machine X will put out machine X+1 or machine 10X, and machine X will no longer be of interest to anyone.
To avoid this dilemma, I have attempted to design an "ideal" computer with very simple rules of operation (requiring, say, only an hour to learn), which also resembles actual machines very closely. There is no reason why a student should be afraid of learning the characteristics of more than one computer; once one machine language has been mastered, others are easily assimilated. Indeed, serious programmers may expect to meet many different machine languages in the course of their careers. So the only remaining disadvantage of a mythical machine is the difficulty of executing any programs written for it. Fortunately, that is not really a problem, because many volunteers have come forward to write simulators for the hypothetical machine. Such simulators are ideal for instructional purposes, since they are even easier to use than a real computer would be.
I have attempted to cite the best early papers in each subject, together with a sampling of more recent work. When referring to the literature, I use standard abbreviations for the names of periodicals, except that the most commonly cited journals are abbreviated as follows:
CACM = Communications of the Association for Computing Machinery
JACM = Journal of the Association for Computing Machinery
Comp. J. = The Computer Journal (British Computer Society)
Math. Comp. = Mathematics of Computation
AMM = American Mathematical Monthly
SICOMP = SIAM Journal on Computing
FOCS = IEEE Symposium on Foundations of Computer Science
SODA = ACM–SIAM Symposium on Discrete Algorithms
STOC = ACM Symposium on Theory of Computing
Crelle = Journal für die reine und angewandte Mathematik
As an example, "CACM 6 (1963), 555–563" stands for the reference given in a preceding paragraph of this preface. I also use " CMath" to stand for the book Concrete Mathematics, which is cited in the introduction to Section 1.2.
Much of the technical content of these books appears in the exercises. When the idea behind a nontrivial exercise is not my own, I have attempted to give credit to the person who originated that idea. Corresponding references to the literature are usually given in the accompanying text of that section, or in the answer to that exercise, but in many cases the exercises are based on unpublished material for which no further reference can be given.
I have, of course, received assistance from a great many people during the years I have been preparing these books, and for this I am extremely thankful. Acknowledgments are due, first, to my wife, Jill, for her infinite patience, for preparing several of the illustrations, and for untold further assistance of all kinds; secondly, to Robert W. Floyd, who contributed a great deal of his time towards the enhancement of this material during the 1960s. Thousands of other people have also provided significant help—it would take another book just to list their names! Many of them have kindly allowed me to make use of hitherto unpublished work. My research at Caltech and Stanford was generously supported for many years by the National Science Foundation and the Office of Naval Research. Addison–Wesley has provided excellent assistance and cooperation ever since I began this project in 1962. The best way I know how to thank everyone is to demonstrate by this publication that their input has led to books that resemble what I think they wanted me to write.
After having spent ten years developing the TeX and METAFONT systems for computer typesetting, I am now able to fulfill the dream that I had when I began that work, by applying those systems to The Art of Computer Programming. At last the entire text of this book has been captured inside my personal computer, in an electronic form that will make it readily adaptable to future changes in printing and display technology. The new setup has allowed me to make literally thousands of improvements that I have been wanting to incorporate for a long time.
In this new edition I have gone over every word of the text, trying to retain the youthful exuberance of my original sentences while perhaps adding some more mature judgment. Dozens of new exercises have been added; dozens of old exercises have been given new and improved answers.
The Art of Computer Programming is, however, still a work in progress. Therefore some parts of this book are headed by an "under construction" icon, to apologize for the fact that the material is not up-to-date. My files are bursting with important material that I plan to include in the final, glorious, fourth edition of Volume 1, perhaps 15 years from now; but I must finish Volumes 4 and 5 first, and I do not want to delay their publication any more than absolutely necessary.
Most of the hard work of preparing the new edition was accomplished by Phyllis Winkler and Silvio Levy, who expertly keyboarded and edited the text of the second edition, and by Jeffrey Oldham, who converted nearly all of the original illustrations to METAPOST format. I have corrected every error that alert readers detected in the second edition (as well as some mistakes that, alas, nobody noticed); and I have tried to avoid introducing new errors in the new material. However, I suppose some defects still remain, and I want to fix them as soon as possible. Therefore I will cheerfully pay $2.56 to the first finder of each technical, typographical, or historical error. The webpage cited on page iv contains a current listing of all corrections that have been reported to me.
D.E.K.
Stanford, California
April 1997
"Things have changed in the past two decades."
—Bill Gates (1995)