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Nikolai Bezroukov. Portraits of Open Source Pioneers
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The Art of Computer Programming - Wikipedia, the free encyclopedia
Finite Semifields and Projective Planes - Donald Knuth's Ph.D. dissertation
Donald E. Knuth, "Algorithmic Themes", in AMS History of Mathematics, Volume 1: A Century of Mathematics in America, AMS, Providence, RI, 1988.
Super Book Deals - More Information Intro to Vol 1 of TAOCP
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 skills that 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:
(a) Some idea of how a stored-program digital computer works; not necessarily the electronics, rather the manner in which instructions can be kept in the machine's memory and successively executed.
(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 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 "non-numerical 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 non-numerical 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 non-numerical techniques, for they are present in the background of numerical programs as well.
The results of research in non-numerical 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, "non-numerical 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 1. Fundamental Algorithms
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 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:
(a) A programmer is greatly influenced by the language in which programs are written; there is an overwhelming tendency to prefer constructions that are simplest in that language, rather than those that are best for the machine. By understanding a machine-oriented language, the programmer will tend to use a much more efficient method; it is much closer to reality.
(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.
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 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.
fas_ci_cle /fas_ ek el / n . . . 1: a small bundle . . . an inflorescence consisting of a compacted cyme less capitate than a glomerule. . . 2: one of the divisions of a book published in parts --P. B. Gove, Webster's Third New International Dictionary (1961)
This is the first of a series of updates that I plan to make available at regular intervals as I continue working toward the ultimate editions of The Art of Computer Programming.
I was inspired to prepare fascicles like this by the example of Charles Dickens, who issued his novels in serial form; he published a dozen installments of Oliver Twist before having any idea what would become of Bill Sikes! I was thinking also of James Murray, who began to publish 350-page portions of the Oxford English Dictionary in 1884, finishing the letter B in 1888 and the letter C in 1895. (Murray died in 1915 while working on the letter T; my task is, fortunately, much simpler than his.)
Unlike Dickens and Murray, I have computers to help me edit the material, so that I can easily make changes before putting everything together in its final form. Although I'm trying my best to write comprehensive accounts that need no further revision, I know that every page brings me hundreds of opportunities to make mistakes and to miss important ideas. My files are bursting with notes about beautiful algorithms that have been discovered, but computer science has grown to the point where I cannot hope to be an authority on all the material I wish to cover. Therefore I need extensive feedback from readers before I can finalize the official volumes.
In other words, I think these fascicles will contain a lot of Good Stuff, and I'm excited about the opportunity to present everything I write to whoever wants to read it, but I also expect that beta-testers like you can help me make it Way Better. As usual, I will gratefully pay a reward of $2.56 to the first person who reports anything that is technically, historically, typographically, or politically incorrect.
Charles Dickens usually published his work once a month, sometimes once a week; James Murray tended to finish a 350-page installment about once every 18 months. My goal, God willing, is to produce two 128-page fascicles per year.Most of the fascicles will represent new material destined for Volumes 4 and higher; but sometimes I will be presenting amendments to one or more of the earlier volumes. For example, Volume 4 will need to refer to topics that belong in Volume 3, but weren't invented when Volume 3 first came out. With luck, the entire work will make sense eventually.
Fascicle Number One is about MMIX, the long-promised replacement for MIX. Thirty-seven years have passed since the MIX computer was designed, and computer architecture has been converging during those years towards a rather different style of machine. Therefore I decided in 1990 to replace MIX with a new computer that would contain even less saturated fat than its predecessor.
Exercise 1.3.1-25 in the first three editions of Volume 1 spoke of an extended MIX called MixMaster, which was upward compatible with the old version. But MixMaster itself has long been hopelessly obsolete. It allowed for several gigabytes of memory, but one couldn't even use it with ASCII code to print lowercase letters. And ouch, its standard conventions for calling subroutines were irrevocably based on self-modifying instructions! Decimal arithmetic and self-modifying code were popular in 1962, but they sure have disappeared quickly as machines have gotten bigger and faster. Fortunately the modern RISC architecture has a very appealing structure, so I've had a chance to design a new computer that is not only up to date but also fun.
Many readers are no doubt thinking, "Why does Knuth replace MIX by another machine instead of just sticking to a high-level programming language? Hardly anybody uses assemblers these days." Such people are entitled to their opinions, and they need not bother reading the machine-language parts of my books. But the reasons for machine language that I gave in the preface to Volume 1, written in the early 1960s, remain valid today:
- One of the principal goals of my books is to show how high-level constructions are actually implemented in machines, not simply to show how they are applied. I explain coroutine linkage, tree structures, random number generation, high-precision arithmetic, radix conversion, packing of data, combinatorial searching, recursion, etc., from the ground up.
- The programs needed in my books are generally so short that their main points can be grasped easily.
- People who are more than casually interested in computers should have at least some idea of what the underlying hardware is like. Otherwise the programs they write will be pretty weird.
- Machine language is necessary in any case, as output of some of the software that I describe.
- Expressing basic methods like algorithms for sorting and searching in machine language makes it possible to carry out meaningful studies of the effects of cache and RAM size and other hardware characteristics (memory speed, pipelining, multiple issue, lookaside buffers, the size of cache blocks, etc.) when comparing different schemes.
Moreover, if I did use a high-level language, what language should it be? In the 1960s I would probably have chosen Algol W; in the 1970s, I would then have had to rewrite my books using Pascal; in the 1980s, I would surely have changed everything to C; in the 1990s, I would have had to switch to C++ and then probably to Java. In the 2000s, yet another language will no doubt be de rigueur. I cannot afford the time to rewrite my books as languages go in and out of fashion; languages aren't the point of my books, the point is rather what you can do in your favorite language. My books focus on timeless truths.
Therefore I will continue to use English as the high-level language in The Art of Computer Programming, and I shall continue to use a low-level language to indicate how machines actually compute. Readers who only want to see algorithms that are already packaged in a plug-in way, using a trendy language, should buy other people's books.
The good news is that programming for MMIX is pleasant and simple. This fascicle presents
1) a programmer's introduction to the machine (replacing Section 1.3.1 of the third edition of Volume 1);
2) the MMIX assembly language (replacing Section 1.3.2);
3) new material on subroutines, coroutines, and interpretive routines (replacing Sections 1.4.1, 1.4.2, and 1.4.3).
Of course, MIX appears in many places throughout the existing editions of Volumes 1--3, and dozens of programs need to be rewritten for MMIX before the next editions of those volumes are ready. Readers who would like to help with this conversion process are encouraged to join the MMIXmasters, a happy group of volunteers based at mmixmasters.sourceforge.net.
The fourth edition of Volume 1 will not be ready until after Volumes 4 and 5 have been completed; therefore two quite different versions of Sections 1.3.1, 1.3.2, 1.4.1, 1.4.2, and 1.4.3 will coexist for several years. In order to avoid potential confusion, I've temporarily assigned "prime numbers" 1.3.1', 1.3.2',1.4.1', 1.4.2', and 1.4.3' to the new material.
I am extremely grateful to all the people who helped me with the design of MMIX. In particular, John Hennessy and Richard L. Sites deserve special thanks for their active participation and substantial contributions. Thanks also to Vladimir Ivanovic for volunteering to be the MMIX grandmaster/webmaster.
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The Last but not Least
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