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Overview
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 lineup.
–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.
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Meet the Author
Donald E. Knuth is known throughout the world for his pioneering work on algorithms and programming techniques, for his invention of the Tex and Metafont systems for computer typesetting, and for his prolific and influential writing. Professor Emeritus of The Art of Computer Programming at Stanford University, he currently devotes full time to the completion of these fascicles and the seven volumes to which they belong.
Read an Excerpt
Procedure for Reading
This Set of Books1. Begin reading this procedure, unless you have already begun to read it. Continue to follow the steps faithfully. (The general form of this procedure and its accompanying flow chart will be used throughout this book.)
2. Read the Notes on the Exercises, on pages xvxvii.
3. Set N equal to 1.
4. Begin reading Chapter N. Do not read the quotations that appear at the beginning of the chapter.
5. Is the subject of the chapter interesting to you? If so, go to step 7; if not, go to step 6.
6. Is N < 2? If not, go to step 16; if so, scan through the chapter anyway. (Chapters 1 and 2 contain important introductory material and also a review of basic programming techniques. You should at least skim over the sections on notation and about MIX.)
7. Begin reading the next section of the chapter; if you have already reached the end of the chapter, however, go to step 16.
8. Is section number marked with "*"? If so, you may omit this section on* first reading (it covers a rather specialized topic that is interesting but not essential); go back to step 7.
9. Are you mathematically inclined? If math is all Greek to you, go to step 11; otherwise proceed to step 10.
10. Check the mathematical derivations made in this section (and report errors to the author). Go to step 12.
11. If the current section is full of mathematical computations, you had better omit reading the derivations. However, you should become familiar with the basic results of the section; they are usually stated near the beginning, or in slanted type right at the very end of the hard parts.
12. Workthe recommended exercises in this section in accordance with the hints given in the Notes on the Exercises (which you read in step 2).
13. After you have worked on the exercises to your satisfaction, check your answers with the answer printed in the corresponding answer section at the rear of the book (if any answer appears for that problem). Also read the answers to the exercises you did not have time to work. Note: In most cases it is reasonable to read the answer to exercise n before working on exercise n + 1, so steps 1213 are usually done simultaneously.
14. Are you tired? If not, go back to step 7.
15. Go to sleep. Then, wake up, and go back to step 7.
16. Increase N by one. If N = 3, 5, 7, 9, 11, or 12, begin the next volume of this set of books.
17. If N is less than or equal to 12, go back to step 4.
18. Congratulations. Now try to get your friends to purchase a copy of Volume 1 and to start reading it. Also, go back to step 3.
Woe be to him that reads but one book.
Le defaut unique de tous les outrages c test d'etre trap longs
Books are a triviality. Life alone Is great
NOTES ON THE EXERCISES
THE EXERCISES in this set of books have been designed for selfstudy as well as classroom study. It is difficult, if not impossible, for anyone to learn a subject purely by reading about it, without applying the information to specific problems and thereby being encouraged to think about what has been read. Furthermore, we all learn best the things that we have discovered for ourselves. Therefore the exercises form a major part of this work; a definite attempt has been made to keep them as informative as possible and to select problems that are enjoyable as well as instructive.
In many books, easy exercises are found mixed randomly among extremely difficult ones. This is sometimes unfortunate because readers like to know in advance how long a problem ought to take_otherwise they may just skip over all the problems. A classic example of such a situation is the book Dynamic Programming by Richard Bellman; this is an important, pioneering work in which a group of problems is collected together at the end of some chapters under the heading "Exercises and Research Problems," with extremely trivial questions appearing in the midst of deep, unsolved problems. It is rumored that someone once asked Dr. Bellman how to tell the exercises apart from the research problems, and he replied, "If you can solve it, it is an exercise; otherwise it's a research problem."
Good arguments can be made for including both research problems and very easy exercises in a book of this kind; therefore, to save the reader from the possible dilemma of determining which are which, rating numbers have been provided to indicate the level of difficulty. These numbers have the following general significance:
Rating Interpretation
The remainder of the rating number divided by 5 indicates the amount of detailed work required. Thus, an exercise rated 24 may take longer to solve than an exercise that is rated 25, but the latter will require more creativity.
The author has tried earnestly to assign accurate rating numbers, but it is difficult for the person who makes up a problem to know Just how formidable it will be for someone else to find a solution; and everyone has more aptitude for certain types of problems than for others. It is hoped that the rating numbers represent a good guess as to the level of difficulty, but they should be taken as general guidelines, not as absolute indicators.
This book has been written for readers with varying degrees of mathematical training and sophistication; as a result, some of the exercises are intended only for the use of more mathematically inclined readers. The rating is preceded by an M if the exercise involves mathematical concepts or motivation to a greater extent than necessary for someone who is primarily interested only in programming the algorithms themselves. An exercise is marked with the letters "HM" if its solution necessarily involves a knowledge of calculus or other higher mathematics not developed in this book. An "HM" designation does not necessarily imply difficulty.
Some exercises are preceded by an arrowhead, "a"; this designates problems that are especially instructive and especially recommended. Of course, no reader/student is expected to work all of the exercises, so those that seem to be the most valuable have been singled out. (This is not meant to detract from the other exercises!) Each reader should at least make an attempt to solve all of the problems whose rating is 10 or less; and the arrows may help to indicate which of the problems with a higher rating should be given priority.
Solutions to most of the exercises appear in the answer section. Please use them wisely; do not turn to the answer until you have made a genuine effort to solve the problem by yourself, or unless you absolutely do not have time to work this particular problem. After getting your own solution or giving the problem a decent try, you may find the answer instructive and helpful. The solution given will often be quite short, and it will sketch the details under the assumption that you have earnestly tried to solve it by your own means first. Sometimes the solution gives less information than was asked; often it gives more. It is quite possible that you may have a better answer than the one published here, or you may have found an error in the published solution; in such a case, the author will be pleased to know the details. Later editions of this book will give the improved solutions together with the solver's name where appropriate.
When working an exercise you may generally use the answers to previous exercises, unless specifically forbidden from doing so. The rating numbers have been assigned with this in mind; thus it is possible for exercise n + 1 to have a lower rating than exercise n + 1 to have a lower rating than exercise n, even though it includes the result of exercise n as a special case.
We can face our problem. We can arrange such facts as we have with order and method.
Table of Contents
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.
2. Information Structures.
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.
Answers to Exercises.
Appendix A. Tables of Numerical Quantities.
1. Fundamental Constants (decimal).
2. Fundamental Constants (octal).
3. Harmonic Numbers, Bernoulli Numbers, Fibonacci Numbers.
Appendix B. Index to Notations.
Index and Glossary. 0201896834T02272003
Preface
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 multivolume 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:
(b) An ability to put the solutions to problems into such explicit terms that a computer can "understand" them. (These machines have no common sense; they do exactly as they are told, no more and no less. This fact is the hardest concept to grasp when one first tries to use a computer.)
(c) Some knowledge of the most elementary computer techniques, such as looping (performing a set of instructions repeatedly), the use of subroutines, and the use of indexed variables.
(d) A little knowledge of common computer jargon—"memory," "registers," "bits," "floating point," "overflow," "software." Most words not defined in the text are given brief definitions in the index at the close of each volume.
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 selfstudy 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 decisionmaking 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:
Chapter 1. Basic Concepts
Chapter 2. Information Structures
Volume 2. Seminumerical Algorithms
Chapter 3. Random Numbers
Chapter 4. Arithmetic
Volume 3. Sorting and Searching
Chapter 5. Sorting
Chapter 6. Searching
Volume 4. Combinatorial Algorithms
Chapter 7. Combinatorial Searching
Chapter 8. Recursion
Volume 5. Syntactical Algorithms
Chapter 9. Lexical Scanning
Chapter 10. Parsing
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 onesemester 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 crossreferences. 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 selfstudy 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 machinelanguage 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 highschool 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 stepbystep description of an algorithm are well known; for a discussion of this, see the article "ComputerDrawn 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 machineoriented language for this purpose. Perhaps many of today's computer experts will disagree with my decision to use a machineoriented 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) Highlevel languages are inadequate for discussing important lowlevel details such as coroutine linkage, random number generation, multiprecision 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 higherlevel programming languages, and it is considerably easier to debug the programs. Indeed, I have rarely used lowlevel 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 machineoriented 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 Xpeople. 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.
Preface to the Third Edition
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 uptodate. 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)
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