Handbook of discrete and combinatorial mathematics

1183 Pages·2000·9.32 MB·English

by  Kenneth H. Rosen

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. A wide variety of references are listed. Reasonable efforts have been made to publish
reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials
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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. COUNTING METHODS
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/Po´lyaCountingFormula — Alan C. Tucker
2.7Mo¨biusInversionCounting — 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. NUMBER THEORY
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
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5. ALGEBRAIC STRUCTURES
— John G. Michaels
5.1AlgebraicModels
5.2Groups
5.3PermutationGroups
5.4Rings
5.5PolynomialRings
5.6Fields
5.7Lattices
5.8BooleanAlgebras
6. LINEAR ALGEBRA
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. DISCRETE PROBABILITY
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. GRAPH THEORY
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
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10. NETWORKS AND FLOWS
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. PARTIALLY ORDERED SETS
11.1BasicPosetConcepts — Graham Brightwell and Douglas B. West
11.2PosetProperties — Graham Brightwell and Douglas B. West
12. COMBINATORIAL DESIGNS
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. DISCRETE AND COMPUTATIONAL GEOMETRY
13.1ArrangementsofGeometricObjects — Ileana Streinu
13.2SpaceFilling — Karoly Bezdek
13.3CombinatorialGeometry — Ja´nos 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. CODING THEORY AND CRYPTOLOGY
— Alfred J. Menezes and
PaulC.vanOorschot
14.1CommunicationSystemsandInformationTheory
14.2BasicsofCodingTheory
14.3LinearCodes
14.4BoundsforCodes
14.5NonlinearCodes
14.6ConvolutionalCodes
14.7BasicsofCryptography
14.8Symmetric-KeySystems
14.9Public-KeySystems
15. DISCRETE OPTIMIZATION
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
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16. THEORETICAL COMPUTER SCIENCE
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. INFORMATION STRUCTURES
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
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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
discretemathematics,focusingontheinformationconsideredessentialtoitsapplication
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.
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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
importantanduseful. Additionalmaterialhasbeenaddedtoensurecomprehensiveness
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.
HowThisBookWasDeveloped
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
ofdiscretemathematics. Theseadvisoryeditorssuggestedchanges,includingthecover-
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.
TheCRCPressSeriesonDiscreteMathematicsandItsApplications
ThisHandbook isdesignedtobeareadyreferencethatcoversmanyimportantdistinct
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
additionalinformationarethevolumesintheCRCPressSeriesonDiscreteMathematics
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
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• 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
Toseeupdatesandtoprovidefeedbackanderratareports,pleaseconsulttheWebpage
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
variousinstitutionswherewework,includingthemanagementofAT&TLaboratoriesfor
theirsupportofthisprojectandforprovidingastimulatingandinterestingatmosphere.
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
withcomputersciencematerialsandcontacts,andAssociateEditorDouglasShierwould
like to thank his wife Joan for her support and understanding throughout the project.
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ADVISORY EDITORIAL BOARD
Andrew Odlyzko — Chief Advisory Editor
AT&T Laboratories
Stephen F. Altschul Frank Harary
National Institutes of Health New Mexico State University
George E. Andrews Alan Hoffman
Pennsylvania State University IBM
Francis T. Boesch Bernard Korte
Stevens Institute of Technology Rheinische Friedrich-Wilhems-Univ.
Ernie Brickell Jeffrey C. Lagarias
Certco AT&T Laboratories
Fan R. K. Chung Carl Pomerance
Univ. of California at San Diego University of Georgia
Charles J. Colbourn Fred S. Roberts
University of Vermont Rutgers University
Stan Devitt Pierre Rosenstiehl
Waterloo Maple Software Centre d’Analyse et de Math. Soc.
Zvi Galil Francis Sullivan
Columbia University IDA
Keith Geddes J. H. VanLint
University of Waterloo Eindhoven University of Technology
Ronald L. Graham Scott Vanstone
Univ. of California at San Diego University of Waterloo
Ralph P. Grimaldi Peter Winkler
Rose-Hulman Inst. of Technology Bell Laboratories
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