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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 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 itnever before told. The story even includes a touch of suspense: two problems that no one has yet been able to solve.
Chapter 7 Combinatorial Searching 1
7.2. Generating All Possibilities 1
7.2.1. Generating Basic Combinatorial Patterns 1
126.96.36.199. Generating all n-tuples 1
188.8.131.52. Generating all permutations 1
184.108.40.206 Generating all combinations 1
220.127.116.11 Generating all partitions 1
18.104.22.168 Generating all set partitions 2
22.214.171.124 Generating all trees 2
126.96.36.199 History and further references 48Answers to Exercises 76Index and Glossary 112
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 188.8.131.52 and 184.108.40.206 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 220.127.116.11 and 18.104.22.168. That sets the stage for the main contents of the present booklet, namely Section 22.214.171.124, which completes the study of basic patterns by discussing how to generate various kinds of tree structures; and Section 126.96.36.199, which completes the story of the preceding subsections by discussing the origins of the concepts and pointing to other sources ofinformation. 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 188.8.131.52-184.108.40.206, 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 220.127.116.11, 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 18.104.22.168 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.
D. E. K.
Posted March 29, 2006
This fascicle can perhaps best be read as a sequel to Knuth's Volume 3, on sorting and searching, where he discusses trees. The fascicle extends that into how does one generate every tree. Of the four fascicles thus published, this might be the skimpiest in terms of current mathematical knowledge. Though to a practising programmer, trees are a vital construct and the book could well have germane analysis. Also, the book contains a distinctive section on the history of combinatorial generation. Knuth delves into this subject while giving a deeper treatment of the maths than one would likely encounter in a popular text directed at a general audience. He cites the I Ching, as well as ancient Indian and Arab manuscripts. The I Ching is notable as it is still in print and likely to be familiar to many. With the publication of this fascicle, the collective set of four would make a respectable book in its own right. However, Knuth is scarcely done yet. We can expect more fascicles, and soon, one might hope. And eventually, a hardcover.Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.