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This multivolume work on the analysis of algorithms has long been recognized as the definitive description of classical computer science.The three complete volumes published to date already comprise a unique and invaluable resource in programming theory and practice. Countless readers have spoken about the profound personal influence of Knuth's writings. Scientists have marveled at the beauty and elegance of his analysis, while practicing programmers have successfully applied his "cookbook" solutions to their day-to-day problems. All have admired Knuth for the breadth, clarity, accuracy, and good humor found in his books.
To begin the fourth and later volumes of the set, and to update parts of the existing three, Knuth has created a series of small books called fascicles, which will be published at regular intervals. Each fascicle will encompass a section or more of wholly new or revised material. Ultimately, the content of these fascicles will be rolled up into the comprehensive, final versions of each volume, and the enormous undertaking that began in 1962 will be complete.Volume 4, Fascicle 3
This fascicle continues Knuth's authoritative chapter on combinatorial algorithms, ultimately to be included in Volume 4 of The Art of Computer Programming. The previous fascicle from Volume 4, which covered the generation of all tuples and permutations, is now complemented by techniques for generating all combinations and partitions. In Knuth's thorough discussion of these two topics, readers will find much that is new, as well as surprisingly rich ties to material in Volumes 1 through 3 and to other aspects of computer science andmathematics. As usual, this fascicle includes a bounty of creative exercises, as well as intriguing challenges posed by yet-unsolved questions.
|Chapter 7||Combinatorial Searching|
|7.2||Generating All Possibilities|
|7.2.1||Generating Basic Combinatorial Patterns|
|126.96.36.199||Generating all n-tuples|
|188.8.131.52||Generating all permutations|
|184.108.40.206||Generating all combinations||1|
|220.127.116.11||Generating all partitions||36|
|18.104.22.168||Generating all set partitions||61|
|Answers to Exercises||87|
|Index and Glossary||144|
In my preface to the first edition, I begged the reader not to draw attention to errors. I now wish I had not done so and am grateful to the few readers who ignored my request.
—Stuart Sutherland, The International Dictionary of Psychology (1996)
This booklet is Fascicle 3 of The Art of Computer Programming, Volume 4: Combinatorial Algorithms. As explained in the preface to Fascicle 1 of Volume 1, I'm circulating the material in this preliminary form because I know that the task of completing Volume 4 will take many years; I can't wait for people to begin reading what I've written so far and to provide valuable feedback.
To put the material in context, this fascicle contains Sections 22.214.171.124, 126.96.36.199, and 188.8.131.52 of a long, long chapter on combinatorial searching. Chapter 7 will eventually fill three volumes (namely Volumes 4A, 4B, and 4C), assuming that I'm able to remain healthy. It will begin with a short review of graph theory, with emphasis on some highlights of significant graphs in The Stanford GraphBase, from which I will be drawing many examples. Then comes Section 7.1, which deals with bitwise manipulation and with algorithms relating to Boolean functions. Section 7.2 is about generating all possibilities, and it begins with Section 7.2.1: Generating Basic Combinatorial Patterns. Details about various useful ways to generate n-tuples appear in Section 184.108.40.206, and the generation of permutations is discussed in Section 220.127.116.11. That sets the stage for the main contents of the present booklet, namely Section 18.104.22.168 (which extends the ideas to combinations of n things taken t at a time); Section 22.214.171.124 (about partitions ofan integer); and Section 126.96.36.199 (about partitions of a set). Then will come Section 188.8.131.52 (about trees) and Section 184.108.40.206 (about the history of combinatorial generation), in Fascicle 4. Section 7.2.2 will deal with backtracking in general. And so it will go on, if all goes well; an outline of the entire Chapter 7 as currently envisaged appears on the taocp webpage that is cited on page ii.
I had great pleasure writing this material, akin to the thrill of excitement that I felt when writing Volume 2 many years ago. As in Volume 2, where I found to my delight that the basic principles of elementary probability theory and number theory arose naturally in the study of algorithms for random number generation and arithmetic, I learned while preparing Section 7.2.1 that the basic principles of elementary combinatorics arise naturally and in a highly motivated way when we study algorithms for combinatorial generation. Thus, I found once again that a beautiful story was "out there" waiting to be told.
For example, in the present booklet we find many of the beautiful patterns formed by combinations, with and without repetition, and how they relate to famous theorems of extremal combinatorics. Then comes my chance to tell the extraordinary story of partitions; indeed, the theory of partitions is one of the nicest chapters in all of mathematics. And in Section 220.127.116.11, a little known triangle of numbers, discovered by C. S. Peirce, turns out to simplify and unify the study of set partitions, another vital topic. Along the way I've included expositions of two mathematical techniques of great importance in the analysis of algorithms: Poisson's summation formula, and the powerful saddle point method. There are games and puzzles too, as in the previous fascicles.
My original intention was to devote far less space to these subjects. But when I saw how fundamental the ideas were for combinatorial studies in general, I knew that I could never be happy unless I covered the basics quite thoroughly. Therefore I've done my best to build a solid foundation of theoretical and practical ideas that will support many kinds of reliable superstructures.
I thank Frank Ruskey for bravely foisting an early draft of this material on college students and for telling me about his classroom experiences. Many other readers have also helped me to check the first drafts; I wish to thank especially George Clements and Svante Janson for their penetrating comments.
I shall happily pay a finder's fee of $2.56 for each error in this fascicle when it is first reported to me, whether that error be typographical, technical, or historical. The same reward holds for items that I forgot to put in the index. And valuable suggestions for improvements to the text are worth 32¢ each. (Furthermore, if you find a better solution to an exercise, I'll actually reward you with immortal glory instead of mere money, by publishing your name in the eventual book.)
Notations that are used here and not otherwise explained can be found in the Index to Notations at the end of Volumes 1, 2, or 3. Those indices point to the places where further information is available. Of course Volume 4 will some day contain its own Index to Notations.
Machine-language examples in all future editions of The Art of Computer Programming will be based on the MMIX computer, which is described in Volume 1, Fascicle 1.
Cross references to yet-unwritten material sometimes appear as '00' in the following pages; this impossible value is a placeholder for the actual numbers to be supplied later.
D. E. K.
Posted October 27, 2005
Only an author as smart and well known in his field as Donald Knuth would have tried this unusual format, which he terms a fascicle. In this third little book, he gives another extensive preview of his eventual fourth volume of 'The Art of Computer Programming'. When it finally appears, this volume is expected to span several books. In the interim, you will have to be content with these fascicles. Even though this book is so slender, it is chock-a-block with tidbits, in the style of the first three volumes. Thus you can find out about a binomial tree, or even an infinite binomial tree. Or see how the Gray binary code also arises in the context of combinations. An elegant aspect of this book is how Knuth ties in the discrete math of combinations with calculus applications. Quite often, these are two different worlds of maths, with different practitioners. Knuth uses the example of the varied properties of Bell numbers. Specifically, the rate at which these grow can be estimated by complex residues and saddle point analysis. Surprising results!
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