Art Of Computer Programming, Volume 4, Fascicle 4,the: Generating All Trees--history Of Combinatorial Generation by Donald E. KnuthArt Of Computer Programming, Volume 4, Fascicle 4,the: Generating All Trees--history Of Combinatorial Generation by Donald E. Knuth

Art Of Computer Programming, Volume 4, Fascicle 4,the: Generating All Trees--history Of…

byDonald E. Knuth

Paperback | February 6, 2006

Pricing and Purchase Info


Earn 128 plum® points

Prices and offers may vary in store


In stock online

Ships free on orders over $25

Not available in stores


Finally, after a wait of more than thirty-five years, the first part of Volume 4 is at last ready for publication. Check out the boxed set that brings together Volumes 1 - 4A in one elegant case, and offers the purchaser a $50 discount off the price of buying the four volumes individually.


The Art of Computer Programming, Volumes 1-4A Boxed Set, 3/e

ISBN: 0321751043 



Art of Computer Programming, Volume 4, Fascicle 4,The: Generating All Trees--History of Combinatorial Generation: Generating All Trees--History of Combinatorial Generation


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 4

This latest fascicle covers the generation of all trees, a basic topic that has surprisingly rich ties to the first three volumes of The Art of Computer Programming. In thoroughly discussing this well-known subject, while providing 124 new exercises, Knuth continues to build a firm foundation for programming. To that same end, this fascicle also covers the history of combinatorial generation. Spanning many centuries, across many parts of the world, Knuth tells a fascinating story of interest and relevance to every artful programmer, much of it never before told. The story even includes a touch of suspense: two problems that no one has yet been able to solve.

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 ful...
Title:Art Of Computer Programming, Volume 4, Fascicle 4,the: Generating All Trees--history Of…Format:PaperbackDimensions:128 pages, 9.5 × 6.4 × 0.35 inPublished:February 6, 2006Publisher:Pearson EducationLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0321335708

ISBN - 13:9780321335708


Read from the Book

I like to work in a variety of fields in order to spread my mistakes more thinly. --Victor Klee (1999) This booklet is Fascicle 4 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 and 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, permutations, combinations, and partitions appear in Sections and That sets the stage for the main contents of the present booklet, namely Section, which completes the study of basic patterns by discussing how to generate various kinds of tree structures; and Section, which completes the story of the preceding subsections by discussing the origins of the concepts and pointing to other sources of information. 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. In fact, I've been looking forward to writing about the generation of trees for a long time, because tree structures have a special place in the hearts of all computer scientists. Although I certainly enjoyed preparing the material about classic combinatorial structures like tuples, permutations, combinations, and partitions in Sections, the truth is that I've saved the best for last: Now it's time for the dessert course. Ever since 1994 I've been giving an annual "Christmas tree lecture" at Stanford University, to talk about the most noteworthy facts about trees that I learned during the current year, and at last I am able to put the contents of those lectures into written form. This topic, like many desserts, is extremely rich, yet immensely satisfying. The theory of trees also ties together a lot of concepts from different aspects of computer programming. And Section, about the history of combinatorial generation, was equally satisfying to the other half of my brain, because it involves poetry, music, religion, philosophy, logic, and intellectual pastimes from many different cultures in many different parts of the world. The roots of combinatorial thinking go very deep, and I can't help but think that I learned a lot about human beings in general as I was putting the pieces of this story together. My original intention was to devote far less space to such subjects. But when I saw how fundamental the ideas were, 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, especially in Section where I was often operating at or beyond the limits of my ability to understand languages other than English. 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:-) 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. Happy reading! D. E. K. Stanford, California June 2005

Table of Contents

Chapter 7 Combinatorial Searching 1

7.2. Generating All Possibilities 1

7.2.1. Generating Basic Combinatorial Patterns 1 Generating all n-tuples 1 Generating all permutations 1 Generating all combinations 1 Generating all partitions 1 Generating all set partitions 2 Generating all trees 2 History and further references 48

Answers to Exercises 76

Index and Glossary 112