Art of Computer Programming, Volume 4B, The: Combinatorial Algorithms
736
Art of Computer Programming, Volume 4B, The: Combinatorial Algorithms
736eBook
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Overview
The Art of Computer Programming is Knuth's multivolume analysis of algorithms. With the addition of this new volume, it continues to be the definitive description of classical computer science.
Volume 4B, the sequel to Volume 4A, extends Knuth's exploration of combinatorial algorithms. These algorithms are of keen interest to software designers because ". . . a single good idea can save years or even centuries of computer time."
The book begins with coverage of Backtrack Programming, together with a set of data structures whose links perform "delightful dances" and are ideally suited to this domain. New techniques for important applications such as optimum partitioning and layout are thereby developed.
Knuth's writing is playful, and he includes dozens of puzzles to illustrate the algorithms and techniques, ranging from popular classics like edge-matching to more recent crazes like sudoku. Recreational mathematicians and computer scientists will not be disappointed!
In the second half of the book, Knuth addresses Satisfiability, one of the most fundamental problems in all of computer science. Innovative techniques developed at the beginning of the twenty-first century have led to game-changing applications, for such things as optimum scheduling, circuit design, and hardware verification. Thanks to these tools, computers are able to solve practical problems involving millions of variables that only a few years ago were regarded as hopeless.
The Mathematical Preliminaries Redux section of the book is a special treat, which presents basic techniques of probability theory that have become prominent since the original "preliminaries" were discussed in Volume 1.
As in every volume of this remarkable series, the book includes hundreds of exercises that employ Knuth's ingenious rating system, making it easy for readers of varying degrees of mathematical training to find challenges suitable to them. Detailed answers are provided to facilitate self-study.
"Professor Donald E. Knuth has always loved to solve problems. In Volume 4B he now promotes two brand new and practical general problem solvers, namely (0) the Dancing Links Backtracking and (1) the SAT Solver. To use them, a problem is defined declaratively (0) as a set of options, or (1) in Boolean formulae. Today's laptop computers, heavily armoured with very high speed processors and ultra large amounts of memory, are able to run either solver for problems having big input data. Each section of Volume 4B contains a multitudinous number of tough exercises which help make understanding surer. Happy reading!" --Eiiti Wada, an elder computer scientist, UTokyo
"Donald Knuth may very well be a great master of the analysis of algorithms, but more than that, he is an incredible and tireless storyteller who always strikes the perfect balance between theory, practice, and fun. [Volume 4B, Combinatorial Algorithms, Part 2] dives deep into the fascinating exploration of search spaces (which is quite like looking for a needle in a haystack or, even harder, to prove the absence of a needle in a haystack), where actions performed while moving forward must be meticulously undone when backtracking. It introduces us to the beauty of dancing links for removing and restoring the cells of a matrix in a dance which is both simple to implement and very efficient." --Christine Solnon, Department of Computer Science, INSA Lyon
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Product Details
| ISBN-13: | 9780137926817 |
|---|---|
| Publisher: | Pearson Education |
| Publication date: | 10/11/2022 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 736 |
| File size: | 64 MB |
| Note: | This product may take a few minutes to download. |
| Age Range: | 18 Years |
About 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 (26 books, 161 papers). Professor Emeritus of The Art of Computer Programming at Stanford University, he currently devotes full time to the completion of his seminal multivolume series on classical computer science, begun in 1962 when he was a graduate student at California Institute of Technology. Professor Knuth is the recipient of numerous awards and honors, including the ACM Turing Award, the Medal of Science presented by President Carter, the AMS Steele Prize for expository writing, and, in November, 1996, the prestigious Kyoto Prize for advanced technology. He lives on the Stanford campus with his wife, Jill.
Read an Excerpt
Chapter 5: Sorting
...Overcoming latency time. Let us consider first the problem of minimizing the delays caused by the fact that the disks aren't always positioned properly when we want to start an I/O command. We can't make the disk spin faster, but we can still apply some tricks that reduce or even eliminate all of the latency time. The addition of more access arms would obviously help, but that would be an expensive hardware modification. Here are some software ideas:
- If we read or write several tracks of a cylinder at a time, we avoid the latency time (and the seek time) on all tracks but the first. In general it is often possible to synchronize the computing time with the disk movement in such a way that a sequence of input/output instructions can be carried out without latency delays.
- Consider the problem of reading half a track of data (Fig. 90): If the read command begins when the heads are at axis A, there is no latency delay, and the total time for reading is just the transmission time, 1/2 x 25ms. If the command begins with the heads at B, we need 1/4 of a revolution for latency and 1/2 for transmission, totaling 3/4 x 25ms. The most interesting case occurs when the heads are initially at C: With proper hardware and software we need not waste 3/4 of a revolution for latency delay. Reading can begin immediately, into the second half of the input buffer; then after a 1/2 x 25ms pause, reading can resume into the first half of the buffer, so that the instruction is completed when axis C is reached again. In a similar manner, we can ensure that the total latency plus transmission time will never exceed the time for one revolution, regardless of the initial position of the disk. The average amount of latency delay is reduced by this scheme from half a revolution to 1/2(1-x2) of a revolution, if we are reading or writing a given fraction x of a track, for 0 < x is less than or equal to 1. When an entire track is being read or written (x=1), this technique eliminates all the latency time.
Let us consider again the example application of Section 5.4.6, sorting 100,000 records of 100 characters each, with a 100,000-character internal memory. The total amount of data to be sorted fills half of a MIXTEC disk. It is usually impossible to read and write simultaneously on a single disk unit; we shall assume that two disks are available, so that reading and writing can overlap each other. For the moment we shall assume, in fact, that the disks are actually drums, containing 4000 tracks of 5000 characters each, with no seek time required.
What sorting algorithm should be used? The method of merging is a fairly natural choice; other methods of internal sorting do not lend themselves so well to a disk implementation, except for the radix techniques of Section 5.2.5. The considerations of Section 5.4.7 show that radix sorting techniques of Section 5.2.5. The considerations of Section 5.4.7 show that radix sorting is usually inferior to merging for general-purpose applications, because the duality theorem of that section applies to disks as well as to tapes. Radix sorting does have a strong advantage, however, when the keys are uniformly distributed and many disks can be used in parallel, because an initial distribution by the most significant digits of the keys will divide the work up into independent subproblems that need no further communication. (See, for example, R.C. Agarwal, SIGMOD Record 25, 2 (June 1996), 240-246.)...
Table of Contents
Preface v
Notes on the Exercises xi
Mathematical Preliminaries Redux 1
Chapter 7: Combinatorial Searching
7.2.2 Backtrack Programming 30
7.2.2.1 Dancing links 65
7.2.2.2 Satisfiability 185
Answers to Exercises 370
Appendix A: Tables of Numerical Quantities 656
Appendix B: Index to Notations 660
Appendix C: Index to Algorithms and Theorems 666
Appendix D: Index to Combinatorial Problems 667
Appendix E: Answers to Puzzles in the Answers 671
Index and Glossary 674
Preface
Cookery is become an art,
a noble science;
cooks are gentlemen.
TITUS LIVIUS, Ab Urbe Condita XXXIX.vi
(Robert Burton, Anatomy of Melancholy 1.2.2.2)
This book forms a natural sequel to the material on information structures in Chapter 2 of Volume 1, because it adds the concept of linearly ordered data to the other basic structural ideas.
The title "Sorting and Searching" may sound as if this book is only for those systems programmers who are concerned with the preparation of general-purpose sorting routines or applications to information retrieval. But in fact the area of sorting and searching provides an ideal framework for discussing a wide variety of important general issues:
- How are good algorithms discovered?
- How can given algorithms and programs be improved?
- How can the efficiency of algorithms be analyzed mathematically?
- How can a person choose rationally between different algorithms for the same task?
- In what senses can algorithms be proved ''best possible''?
- How does the theory of computing interact with practical considerations?
- How can external memories like tapes, drums, or disks be used efficiently with large databases?
Indeed, I believe that virtually every important aspect of programming arises somewhere in the context of sorting or searching!
This volume comprises Chapters 5 and 6 of the complete series. Chapter 5 is concerned with sorting into order; this is a large subject that has been divided chiefly into two parts, internal sorting and external sorting. There also are supplementary sections, which develop auxiliarytheories about permutations (Section 5.1) and about optimum techniques for sorting (Section 5.3). Chapter 6 deals with the problem of searching for specified items in tables or files; this is subdivided into methods that search sequentially, or by comparison of keys, or by digital properties, or by hashing, and then the more difficult problem of secondary key retrieval is considered. There searching related to sorting is a surprising amount of interplay between both chapters, with strong analogies tying the topics together. Two important varieties of information structures are also discussed, in addition to those considered in Chapter 2, namely priority queues (Section 5.2.3) and linear lists represented as balanced trees (Section 6.2.3).
Like Volumes 1 and 2, this book includes a lot of material that does not appear in other publications. Many people have kindly written to me about their ideas, or spoken to me about them, and I hope that I have not distorted the material too badly when I have presented it in my own words.
I have not had time to search the patent literature systematically; indeed, I decry the current tendency to seek patents on algorithms (see Section 5.4.5). If somebody sends me a copy of a relevant patent not presently cited in this book, I will dutifully refer to it in future editions. However, I want to encourage people to continue the centuries-old mathematical tradition of putting newly discovered algorithms into the public domain. There are better ways to earn a living than to prevent other people from making use of one's contributions to computer science.
Before I retired from teaching, I used this book as a text for a student's second course in data structures, at the junior-to-graduate level, omitting most of the mathematical material. I also used the mathematical portions of this book as the basis for graduate-level courses in the analysis of algorithms, emphasizing especially Sections 5.1, 5.2.2, 6.3, and 6.4. A graduate-level course on concrete computational complexity could also be based on Sections 5.3, and 5.4.4, together with Sections 4.3.3, 4.6.3, and 4.6.4 of Volume 2.
For the most part this book is self-contained, except for occasional discussions relating to the MIX computer explained in Volume 1. Appendix B MIX computer contains a summary of the mathematical notations used, some of which are a little different from those found in traditional mathematics books.
Preface to the Second Edition
This new edition matches the third editions of Volumes 1 and 2, in which I have been able to celebrate the completion of Tex and MF by applying those systems to the publications they were designed for.
The conversion to electronic format has given me the opportunity to go over every word of the text and every punctuation mark. I've tried 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. Changes appear everywhere, but most significantly in Sections 5.1.4 (about permutations and tableaux), 5.3 (about optimum sorting), 5.4.9 (about disk sorting), 6.2.2 (about entropy), 6.4 (about universal hashing), and 6.5 (about multidimensional trees and tries).
The Art of Computer Programming is, however, still a work in progress. Research on sorting and searching continues to grow at a phenomenal rate. 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. For example, if I were teaching an undergraduate class on data structures today, I would surely discuss randomized structures such as treaps at some length; but at present, I am only able to cite the principal papers on the subject, and to announce plans for a future Section 6.2.5 (see page 6.2.5). My files are bursting with important material that I plan to include in the final, glorious, third edition of Volume 3, perhaps 17 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.
I am enormously grateful to the many hundreds of people who have helped me to gather and refine this material during the past 35 years. Most of the hard work of preparing the new edition was accomplished by Phyllis Winkler (who put the text of the first edition into Tex form), by Silvio Levy (who edited it extensively and helped to prepare several dozen illustrations), and by Jeffrey Oldham (who converted more than 250 of the original illustrations to METAPOST format). The production staff at Addison Wesley has also been extremely helpful, as usual.
D. E. K.Stanford, California
February 1998
There are certain common Privileges of a Writer,
the Benefit whereof, I hope, there will be no Reason to doubt;
Particularly, that where I am not understood, it shall be concluded,
that something very useful and profound is coucht underneath.
JONATHAN SWIFT, Tale of a Tub, Preface (1704)