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About This eBook ePUB is an open, industry-standard format for eBooks. However, support of ePUB and its many features varies across reading devices and applications. Use your device or app settings to customize the presentation to your liking. Settings that you can customize often include font, font size, single or double column, landscape or portrait mode, and figures that you can click or tap to enlarge. For additional information about the settings and features on your reading device or app, visit the device manufacturer’s Web site. Many titles include programming code or configuration examples. To optimize the presentation of these elements, view the eBook in single-column, landscape mode and adjust the font size to the smallest setting. In addition to presenting code and configurations in the reflowable text format, we have included images of the code that mimic the presentation found in the print book; therefore, where the reflowable format may compromise the presentation of the code listing, you will see a “Click here to view code image” link. Click the link to view the print-fidelity code image. To return to the previous page viewed, click the Back button on your device or app. The Art of Computer Programming The Art of Computer Programming Volume 4B / Combinatorial Algorithms, Part 2 Donald E. Knuth Stanford University Boston • Columbus • New York • San Francisco • Amsterdam • Cape Town Dubai • London • Madrid • Milan • Munich • Paris • Montréal • Toronto • Delhi • Mexico City São Paulo • Sydney • Hong Kong • Seoul • Singapore • Taipei • Tokyo Lyrics have been quoted on page 65 from the songs “Mississippi Mud,” written by Harry Barris and James Cavanaugh, and “Pick Yourself Up,” written by Dorothy Fields and Jerome Kern. Used by permission of Shapiro, Bernstein & Co, Inc. The author and publisher have taken care in the preparation of this book, but make no expressed or implied warranty of any kind and assume no responsibility for errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of the use of the information or programs contained herein. For government sales inquiries, please contact [email protected] For questions about sales outside the U.S., please contact [email protected] Visit us on the Web: informit.com/aw Library of Congress Cataloging-in-Publication Data Knuth, Donald Ervin, 1938- The art of computer programming / Donald Ervin Knuth. xviii, 714 p. 24 cm. Includes bibliographical references and index. Contents: v. 1. Fundamental algorithms. -- v. 2. Seminumerical algorithms. -- v. 3. Sorting and searching. -- v. 4a. Combinatorial algorithms, part 1. -- v. 4b. Combinatorial algorithms, part 2. Contents: v. 4b. Combinatorial algorithms, part 2. ISBN 978-0-201-89683-1 (v. 1, 3rd ed.) ISBN 978-0-201-89684-8 (v. 2, 3rd ed.) ISBN 978-0-201-89685-5 (v. 3, 2nd ed.) ISBN 978-0-201-03804-0 (v. 4a) ISBN 978-0-201-03806-4 (v. 4b) 1. Electronic digital computers--Programming. 2. Computer algorithms. I. Title. QA76.6.K64 1997 005.1--DC21 97-2147 Internet page http://www-cs-faculty.stanford.edu/~knuth/taocp.xhtml contains current information about this book and related books. See also for http://www-cs-faculty.stanford.edu/~knuth/sgb.xhtml information about The Stanford GraphBase, including downloadable software for dealing with the graphs used in many of the examples. And see for basic http://www-cs-faculty.stanford.edu/~knuth/mmix.xhtml information about the computer. MMIX Copyright © 2023 by Pearson Education, Inc. All rights reserved. This publication is protected by copyright, and permission must be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions, request forms, and the appropriate contacts with the Pearson Education Global Rights & Permissions Department, please visit . www.pearson.com/permissions/ ISBN-13 978-0-201-03806-4 ISBN-10 0-201-03806-4 First printing, October 2022 First digital release, October 2022 Preface Begin at the beginning, and do not allow yourself to gratify a mere idle curiosity by dipping into the book, here and there. This would very likely lead to your throwing it aside, with the remark “This is much too hard for me!,” and thus losing the chance of adding a very large item to your stock of mental delights. — LEWIS CARROLL, in Symbolic Logic (1896) Combinatorial algorithms are the methods that allow us to cope with problems that involve zillions of cases. The explosive growth in the knowledge of such techniques has meant that several volumes are needed to describe them. Thus my original plan to devote Volume 4 of The Art of Computer Programming to combinatorial algorithms has morphed into a plan to prepare Volumes 4A, 4B, and so on. This book is the second of that series, a sequel to Volume 4A. In the preface to Volume 4A I explained why I was captivated by combinatorial algorithms soon after I fell in love with computers. “The art of writing such programs is especially important and appealing because a single good idea can save years or even centuries of computer time.” Chapter 7 began in Volume 4A with a short review of graph theory and a longer discussion of “Zeros and Ones” (Section 7.1). That volume concluded with Section 7.2.1, “Generating Basic Combinatorial Patterns,” which was the first part of Section 7.2, “Generating All Possibilities.” Now the story continues, with the opening parts of Section 7.2.2, “Backtrack Programming.” Backtracking is the name for an important body of techniques that have been a mainstay of combinatorial algorithms since the beginning. More than a third of this book is devoted to Section 7.2.2.1, which explores data structures whose links perform delightful dances. Such structures are ideally suited to backtrack programming in general, and to the “exact cover problem” (XC) in particular. The XC problem, also known as “set partitioning,” essentially asks for all ways to cover a set of items, by choosing appropriate subsets of items called options. Dozens of important applications turn out to be special cases of XC, and the method of choice for such problems is often to use dancing links. While writing this material I learned to my surprise that an apparently innocuous extension of the classical XC problem leads to an enormous increase in the number of significant special cases. This extended problem, called XCC (for “exact covering with colors”), allows some of the items to receive various colors. Colored items are allowed to be covered by many different options, as long as the colors are compatible. Spoiler alert: With dancing links, we can solve XCC problems almost as easily as XC problems! Therefore I believe that the study of XCC solvers, now in its infancy, is destined to become quite important, and I’ve done my best to introduce the subject here. There also are related methods for an even more general class of problems called MCC (“multiple covering with colors”), and for finding XCC solutions of minimum cost. If you turn to a random page of Section 7.2.2.1, chances are good that you’ll find some sort of puzzle being discussed. The reason is that puzzles are by far the best means I know to illustrate the algorithms and techniques that are being introduced here. The point of a puzzle is easily grasped; and the fact that an extraordinary number of quite different puzzles all turn out to be special cases of XCC and MCC is significant in itself. Indeed, it becomes clear that the same ideas will solve many complex and harder-to-explain problems of the “real world.”