HANDBOOK OF DISCRETE AND COMBINATORIAL UTHEMATICS KENNETH H. ROSEN AT&T Laboratories Editor-in-Chief JOHN G. MICHAELS SUNY Brockport Project Editor JONATHAN L. GROSS Columbia University Associate Editor JERROLD W. GROSSMAN Oakland University Associate Editor DOUGLAS R SHIER Clemson University Associate Editor CRC Press Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Publication Data Handbook of discrete and combinatorial mathematics / Kenneth H. Rosen, editor in chief, John G. Michaels, project editor [et al.]. p. cm. Includes bibliographical references and index. ISBN 0-8493-0149-1 (alk. paper) 1. Combinatorial analysis-Handbooks, manuals, etc. 2. Computer science-Mathematics-Handbooks, manuals, etc. I. Rosen, Kenneth H. II. Michaels, John G. QAl64.H36 1999 5 I I .‘6—dc21 99-04378 This book contains information obtained from authentic and highIy regarded sources. Reprinted materia1 is quoted with permission, and sources are indicated. 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For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2000 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-0149-1 Library of Congress Card Number 99-04378 Printed in the United States of America 4 5 6 7 8 9 IO 11 12 13 Printed on acid-free paper CONTENTS 1.FOUNDATIONS 1.1PropositionalandPredicateLogic— Jerrold W. Grossman 1.2SetTheory— Jerrold W. Grossman 1.3Functions— Jerrold W. Grossman 1.4Relations— John G. Michaels 1.5ProofTechniques— Susanna S. Epp 1.6AxiomaticProgramVerification— David Riley 1.7Logic-BasedComputerProgrammingParadigms— Mukesh Dalal 2.COUNTINGMETHODS 2.1SummaryofCountingProblems— John G. Michaels 2.2BasicCountingTechniques— Jay Yellen 2.3PermutationsandCombinations— Edward W. Packel 2.4Inclusion/Exclusion— Robert G. Rieper 2.5Partitions— George E. Andrews 2.6Burnside/P´olyaCountingFormula— Alan C. Tucker 2.7M¨obiusInversionCounting— Edward A. Bender 2.8YoungTableaux— Bruce E. Sagan 3.SEQUENCES 3.1SpecialSequences— Thomas A. Dowling and Douglas R. Shier 3.2GeneratingFunctions— Ralph P. Grimaldi 3.3RecurrenceRelations— Ralph P. Grimaldi 3.4FiniteDifferences— Jay Yellen 3.5FiniteSumsandSummation— Victor S. Miller 3.6AsymptoticsofSequences— Edward A. Bender 3.7MechanicalSummationProcedures— Kenneth H. Rosen 4.NUMBERTHEORY 4.1BasicConcepts— Kenneth H. Rosen 4.2GreatestCommonDivisors— Kenneth H. Rosen 4.3Congruences— Kenneth H. Rosen 4.4PrimeNumbers— Jon F. Grantham and Carl Pomerance 4.5Factorization— Jon F. Grantham and Carl Pomerance 4.6ArithmeticFunctions— Kenneth H. Rosen 4.7PrimitiveRootsandQuadraticResidues— Kenneth H. Rosen 4.8DiophantineEquations— Bart E. Goddard 4.9DiophantineApproximation— Jeff Shalit 4.10QuadraticFields— Kenneth H. Rosen c  2000 by CRC Press LLC 5.ALGEBRAICSTRUCTURES— John G. Michaels 5.1AlgebraicModels 5.2Groups 5.3PermutationGroups 5.4Rings 5.5PolynomialRings 5.6Fields 5.7Lattices 5.8BooleanAlgebras 6.LINEARALGEBRA 6.1VectorSpaces— Joel V. Brawley 6.2LinearTransformations— Joel V. Brawley 6.3MatrixAlgebra— Peter R. Turner 6.4LinearSystems— Barry Peyton and Esmond Ng 6.5Eigenanalysis— R. B. Bapat 6.6CombinatorialMatrixTheory— R. B. Bapat 7.DISCRETEPROBABILITY 7.1FundamentalConcepts— Joseph R. Barr 7.2IndependenceandDependence— Joseph R. Barr 435 7.3RandomVariables— Joseph R. Barr 7.4DiscreteProbabilityComputations— Peter R. Turner 7.5RandomWalks— Patrick Jaillet 7.6SystemReliability— Douglas R. Shier 7.7Discrete-TimeMarkovChains— Vidyadhar G. Kulkarni 7.8QueueingTheory— Vidyadhar G. Kulkarni 7.9Simulation— Lawrence M. Leemis 8.GRAPHTHEORY 8.1IntroductiontoGraphs— Lowell W. Beineke 8.2GraphModels— Jonathan L. Gross 8.3DirectedGraphs— Stephen B. Maurer 8.4Distance,Connectivity,Traversability— Edward R. Scheinerman 8.5GraphInvariantsandIsomorphismTypes— Bennet Manvel 8.6GraphandMapColoring— Arthur T. White 8.7PlanarDrawings— Jonathan L. Gross 8.8TopologicalGraphTheory— Jonathan L. Gross 8.9EnumeratingGraphs— Paul K. Stockmeyer 8.10AlgebraicGraphTheory— Michael Doob 8.11AnalyticGraphTheory— Stefan A. Burr 8.12Hypergraphs— Andreas Gyarfas 9.TREES 9.1CharacterizationsandTypesofTrees— Lisa Carbone 9.2SpanningTrees— Uri Peled 9.3EnumeratingTrees— Paul Stockmeyer c  2000 by CRC Press LLC 10.NETWORKSANDFLOWS 10.1MinimumSpanningTrees— J. B. Orlin and Ravindra K. Ahuja 10.2Matchings— Douglas R. Shier 10.3ShortestPaths— J. B. Orlin and Ravindra K. Ahuja 10.4MaximumFlows— J. B. Orlin and Ravindra K. Ahuja 10.5MinimumCostFlows— J. B. Orlin and Ravindra K. Ahuja 10.6CommunicationNetworks— David Simchi-Levi and Sunil Chopra 10.7DifficultRoutingandAssignmentProblems— Bruce L. Golden and Bharat K. Kaku 10.8NetworkRepresentationsandDataStructures— Douglas R. Shier 11.PARTIALLYORDEREDSETS 11.1BasicPosetConcepts— Graham Brightwell and Douglas B. West 11.2PosetProperties— Graham Brightwell and Douglas B. West 12.COMBINATORIALDESIGNS 12.1BlockDesigns— Charles J. Colbourn and Jeffrey H. Dinitz 12.2SymmetricDesigns&FiniteGeometries— Charles J. Colbourn and Jeffrey H. Dinitz 12.3LatinSquaresandOrthogonalArrays— Charles J. Colbourn and Jeffrey H. Dinitz 12.4Matroids— James G. Oxley 13.DISCRETEANDCOMPUTATIONALGEOMETRY 13.1ArrangementsofGeometricObjects— Ileana Streinu 13.2SpaceFilling— Karoly Bezdek 13.3CombinatorialGeometry— J´anos Pach 13.4Polyhedra— Tamal K. Dey 13.5AlgorithmsandComplexityinComputationalGeometry— Jianer Chen 13.6GeometricDataStructuresandSearching— Dina Kravets 853 13.7ComputationalTechniques— Nancy M. Amato 13.8ApplicationsofGeometry— W. Randolph Franklin 14.CODINGTHEORYANDCRYPTOLOGY— Alfred J. Menezes and Paul C. van Oorschot 14.1CommunicationSystemsandInformationTheory 14.2BasicsofCodingTheory 14.3LinearCodes 14.4BoundsforCodes 14.5NonlinearCodes 14.6ConvolutionalCodes 14.7BasicsofCryptography 14.8Symmetric-KeySystems 14.9Public-KeySystems 15.DISCRETEOPTIMIZATION 15.1LinearProgramming— Beth Novick 15.2LocationTheory— S. Louis Hakimi 15.3PackingandCovering— Sunil Chopra and David Simchi-Levi 15.4ActivityNets— S. E. Elmaghraby 15.5GameTheory— Michael Mesterton-Gibbons 15.6Sperner’sLemmaandFixedPoints— Joseph R. Barr c  2000 by CRC Press LLC 16.THEORETICALCOMPUTERSCIENCE 16.1ComputationalModels— Jonathan L. Gross 16.2Computability— William Gasarch 16.3LanguagesandGrammars— Aarto Salomaa 16.4AlgorithmicComplexity— Thomas Cormen 16.5ComplexityClasses— Lane Hemaspaandra 16.6RandomizedAlgorithms— Milena Mihail 17.INFORMATIONSTRUCTURES 17.1AbstractDatatypes— Charles H. Goldberg 17.2ConcreteDataStructures— Jonathan L. Gross 17.3SortingandSearching— Jianer Chen 17.4Hashing— Viera Krnanova Proulx 17.5DynamicGraphAlgorithms— Joan Feigenbaum and Sampath Kannan BIOGRAPHIES— Victor J. Katz c  2000 by CRC Press LLC PREFACE The importance of discrete and combinatorial mathematics has increased dramatically within the last few years. The purpose of the Handbook of Discrete and Combinatorial Mathematics is to provide a comprehensive reference volume for computer scientists, engineers, mathematicians, and others, such as students, physical and social scientists, and reference librarians, who need information about discrete and combinatorial math- ematics. This book is the first resource that presents such information in a ready-reference form designed for use by all those who use aspects of this subject in their work or studies. The scope of this book includes the many areas generally considered to be parts of discrete mathematics, focusing on the information considered essential to its application in computer science and engineering. Some of the fundamental topic areas covered include: logic and set theory graph theory enumeration trees integer sequences network sequences recurrence relations combinatorial designs generating functions computational geometry number theory coding theory and cryptography abstract algebra discrete optimization linear algebra automata theory discrete probability theory data structures and algorithms. Format The material in the Handbook is presented so that key information can be located and used quickly and easily. Each chapter includes a glossary that provides succinct definitions of the most important terms from that chapter. Individual topics are cov- ered in sections and subsections within chapters, each of which is organized into clearly identifiable parts: definitions, facts, and examples. The definitions included are care- fully crafted to help readers quickly grasp new concepts. Important notation is also highlighted in the definitions. Lists of facts include: • information about how material is used and why it is important • historical information • key theorems • the latest results • the status of open questions • tables of numerical values, generally not easily computed • summary tables • key algorithms in an easily understood pseudocode • information about algorithms, such as their complexity • major applications • pointers to additional resources, including websites and printed material. c  2000 by CRC Press LLC Facts are presented concisely and are listed so that they can be easily found and un- derstood. Extensive crossreferences linking parts of the handbook are also provided. Readers who want to study a topic further can consult the resources listed. The material in the Handbook has been chosen for inclusion primarily because it is important and useful. Additional material has been added to ensure comprehensiveness so that readers encountering new terminology and concepts from discrete mathematics in their explorations will be able to get help from this book. Examples are provided to illustrate some of the key definitions, facts, and algorithms. Some curious and entertaining facts and puzzles that some readers may find intriguing are also included. Each chapter of the book includes a list of references divided into a list of printed resources and a list of relevant websites. How This Book Was Developed The organization and structure of the Handbook were developed by a team which in- cluded the chief editor, three associate editors, the project editor, and the editor from CRC Press. This team put together a proposed table of contents which was then ana- lyzed by members of a group of advisory editors, each an expert in one or more aspects of discrete mathematics. These advisory editors suggested changes, including the cover- age of additional important topics. Once the table of contents was fully developed, the individual sections of the book were prepared by a group of more than 70 contributors from industry and academia who understand how this material is used and why it is important. Contributors worked under the direction of the associate editors and chief editor, with these editors ensuring consistency of style and clarity and comprehensive- ness in the presentation of material. Material was carefully reviewed by authors and our team of editors to ensure accuracy and consistency of style. The CRC Press Series on Discrete Mathematics and Its Applications This Handbook is designed to be a ready reference that covers many important distinct topics. People needing information in multiple areas of discrete and combinatorial mathematics need only have this one volume to obtain what they need or for pointers to where they can find out more information. Among the most valuable sources of additional information are the volumes in the CRC Press Series on Discrete Mathematics and Its Applications. This series includes both Handbooks, which are ready references, and advanced Textbooks/Monographs. More detailed and comprehensive coverage in particular topic areas can be found in these individual volumes: Handbooks • The CRC Handbook of Combinatorial Designs • Handbook of Discrete and Computational Geometry • Handbook of Applied Cryptography Textbooks/Monographs • Graph Theory and its Applications • Algebraic Number Theory • Quadratics c  2000 by CRC Press LLC • Design Theory • Frames and Resolvable Designs: Uses, Constructions, and Existence • Network Reliability: Experiments with a Symbolic Algebra Environment • Fundamental Number Theory with Applications • Cryptography: Theory and Practice • Introduction to Information Theory and Data Compression • Combinatorial Algorithms: Generation, Enumeration, and Search Feedback To see updates and to provide feedback and errata reports, please consult the Web page for this book. This page can be accessed by first going to the CRC website at http://www.crcpress.com and then following the links to the Web page for this book. Acknowledgments First and foremost, we would like to thank the original CRC editor of this project, Wayne Yuhasz, who commissioned this project. We hope we have done justice to his original vision of what this book could be. We would also like to thank Bob Stern, who has served as the editor of this project for his continued support and enthusiasm for this project. We would like to thank Nora Konopka for her assistance with many aspects in the development of this project. Thanks also go to Susan Fox, for her help with production of this book at CRC Press. We would like to thank the many people who were involved with this project. First, we would like to thank the team of advisory editors who helped make this reference relevant, useful, unique, and up-to-date. We also wish to thank all the people at the various institutions where we work, including the management of AT&T Laboratories for their support of this project and for providing a stimulating and interesting atmosphere. Project Editor John Michaels would like to thank his wife Lois and daughter Margaret for their support and encouragement in the development of the Handbook. Associate Editor Jonathan Gross would like to thank his wife Susan for her patient support, Associate Editor Jerrold Grossman would like to thank Suzanne Zeitman for her help with computer science materials and contacts, and Associate Editor Douglas Shier would like to thank his wife Joan for her support and understanding throughout the project. c  2000 by CRC Press LLC ADVISORY EDITORIAL BOARD Andrew Odlyzko — Chief Advisory Editor AT&T Laboratories Stephen F. Altschul National Institutes of Health George E. Andrews Pennsylvania State University Francis T. Boesch Stevens Institute of Technology Ernie Brickell Certco FanR.K.Chung Univ. of California at San Diego Charles J. Colbourn University of Vermont Stan Devitt Waterloo Maple Software Zvi Galil Columbia University Keith Geddes University of Waterloo Ronald L. Graham Univ. of California at San Diego Ralph P. Grimaldi Rose-Hulman Inst. of Technology Frank Harary New Mexico State University Alan Hoffman IBM Bernard Korte Rheinische Friedrich-Wilhems-Univ. Jeffrey C. Lagarias AT&T Laboratories Carl Pomerance University of Georgia Fred S. Roberts Rutgers University Pierre Rosenstiehl Centre d’Analyse et de Math. Soc. Francis Sullivan IDA J. H. Van Lint Eindhoven University of Technology Scott Vanstone University of Waterloo Peter Winkler Bell Laboratories c  2000 by CRC Press LLC [...]... to mathematics and astronomy, and entered the ministry In 1682 be began c 2000 by CRC Press LLC to lecture at the University of Basil in natural philosophy and mechanics He became professor at the University of Basel in 1687, and remained there until his death His research included the areas of the calculus of variations, probability, and analytic geometry His most well-known work is Ars Conjectandi,... general of Alexander the Great who became ruler of Egypt after Alexander’s death in 323 B.C.E Leonhard Euler (1707–1783) was born in Basel, Switzerland and became one of the earliest members of the St Petersburg Academy of Sciences He was the most prolific mathematician of all time, making contributions to virtually every area of the subject His series of analysis texts established many of the notations and. .. Mathematical Association of America, 1983 V J Katz, History of Mathematics, an Introduction, 2nd ed., Addison-Wesley, 1998 Web Resource: http://www-groups.dcs.st -and. ac.uk/~history (The MacTutor History of Mathematics archive.) c 2000 by CRC Press LLC 1 FOUNDATIONS 1.1 Propositional and Predicate Logic 1.1.1 Propositions and Logical Operations 1.1.2 Equivalences, Identities, and Normal Forms 1.1.3 Predicate... FORTRAN He was a developer of ALGOL, using the Backus-Naur form for the syntax of the language He received the National Medal of Science in 1974 and the Turing Award in 1977 Abu-l-’Abbas Ahmad ibn Muhammad ibn al-Banna al-Marrakushi (1256– 1321) was an Islamic mathematician who lived in Marrakech in what is now Morocco Ibn al-Banna developed the first known proof of the basic combinatorial formulas, beginning... determining the number of partitions of an integer n into m distinct parts, each of which is in a given set A of distinct positive integers And in a paper of 1782, he even posed the problem of the existence of a pair of orthogonal latin squares: If there are 36 officers, one of each of six ranks from each of six different regiments, can they be arranged in a square in such a way that each row and column contains... book A Course in Pure Mathematics revolutionized mathematics teaching, and his book A Mathematician’s Apology gives his view of what mathematics is and the value of its study c 2000 by CRC Press LLC Ab¯ ’Al¯ al-Hasan ibn al-Haytham (Alhazen) (965–1039) was one of the most u i influential of Islamic scientists He was born in Basra (now in Iraq) but spent most of his life in Egypt, after he was invited to... University of Lw´w and taught at the o University of Lw´w, the University of Warsaw, and the Royal Irish Academy A o logician, he worked in the area of many-valued logic, writing papers on three-valued and m-valued logics, He is best known for the parenthesis-free notation he developed for propositions, called Polish notation Percy Alexander MacMahon (1854–1929) was born into a British army family and joined... movement in pre-World War I Russia and often criticized publicly the actions of state authorities In 1913, when as a member of the Academy c 2000 by CRC Press LLC of Sciences he was asked to participate in the pompous ceremonies celebrating the 300th anniversary of the Romanov dynasty, he instead organized a celebration of the 200th anniversary of Jacob Bernoulli’s publication of the Law of Large Numbers... Ptolemy III Euergetes in Alexandria and became chief librarian at Alexandria He is recognized as the foremost scholar of his time and wrote in many areas, including number theory (his sieve for obtaining primes) and geometry He introduced the concepts of meridians of longitude and parallels of latitude and used these to measure distances, including an estimation of the circumference of the earth Paul Erd˝s... 1930 that any non-planar graph must contain a copy of one of two particularly simple non-planar graphs Joseph Louis Lagrange (1736–1813) was born in Turin into a family of French descent He was attracted to mathematics in school and at the age of 19 became a mathematics professor at the Royal Artillery School in Turin At about the same time, having read a paper of Euler’s on the calculus of variations, . LLC PREFACE The importance of discrete and combinatorial mathematics has increased dramatically within the last few years. The purpose of the Handbook of Discrete and Combinatorial Mathematics is to provide. detailed and comprehensive coverage in particular topic areas can be found in these individual volumes: Handbooks • The CRC Handbook of Combinatorial Designs • Handbook of Discrete and Computational. bibliographical references and index. ISBN 0-8 49 3-0 14 9-1 (alk. paper) 1. Combinatorial analysis-Handbooks, manuals, etc. 2. Computer science -Mathematics- Handbooks, manuals, etc. I. Rosen, Kenneth