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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?

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 .Ô6Ñdc21 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 abssume responsibility for the validity of all materials

or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any fborm or by any means, electronic or mechanical,

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© 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

1.6AxiomaticProgramVerication - ?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

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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

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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

10.7DifcultRoutingandAssignmentProblems - ?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. va nOorschot

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

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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

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2000 by CRC Press LLC

PREFACE

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Acknowledgments

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