Algorithmic Problem Solving

426 Pages·2011·14.14 MB·English

by  Roland C. Backhouse

Algorithmic
Problem Solving
Roland Backhouse
University ofNottingham
@WILEY
AJohnWiley&Sons.Ltd..Publication
Thiseditionfirstpublished2011
©2011JohnWiley86SonsLtd.
Registeredoffice
JohnWiley8:SonsLtd.,TheAtrium,SouthernGate,Chichester,WestSussex,P019SSQ,UnitedKingdom
Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapplyfor
permissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com.
TherightsofRolandBackhousetobeidentifiedastheauthorofthisworkhasbeenassertedinaccordancewith
theCopyright,DesignsandPatentsAct1988.
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmitted,in
anyformorbyanymeans,electronic,mechanical,photocopying,recordingorotherwise,exceptaspermittedby
theUKCopyright,DesignsandPatentsAct1988,withoutthepriorpermissionofthepublisher.
Wileyalsopublishesitsbooksinavarietyofelectronicformats.Somecontentthatappearsinprintmaynotbe
availableinelectronicbooks.
Designationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarks.Allbrandnames
andproductnamesusedinthisbookaretradenames,servicemarks,trademarksorregisteredtrademarksof
theirrespectiveowners.Thepublisherisnotassociatedwithanyproductorvendormentionedinthisbook.This
publicationisdesignedtoprovideaccurateandauthoritativeinformationinregardtothesubjectmattercovered.
Itissoldontheunderstandingthatthepublisherisnotengagedinrenderingprofessionalservices.Ifprofessional
adviceorotherexpertassistanceisrequired,theservicesofacompetentprofessionalshouldbesought.
LibraryofCongressCataloging-in-PublicationData
Backhouse,RolandC.,1948-
Algorithmicproblemsolving/RolandBackhouse.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-68453-5(pbk.:alk.paper)
1.Computeralgorithms.2.Problemsolving.I.Title.
QA76.9.A43B342011
005.1-ch3
2011022716
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
Setin9/12SabonbyLaserwordsPrivateLtd.,Chennai,India.
PrintedinGreatBritainbyT]International,Padstow,Cornwall

CONTENTS
CHAPTER 4 - Games
4.1 Matchstick Games
4.2 WinningStrategies
4.2.1 Assumptions
4.2.2 LabellingPositions
4.2.3 FormulatingRequirements
4.3 Subtraction-SetGames
4.4 SumsofGames
4.4.1 A Simple SumGame
4.4.2 Maintain Symmetry!
4.4.3 More Simple Sums
4.4.4 EvaluatingPositions
4.4.5 UsingtheMexFunction
4.5 Summary
4.6 BibliographicRemarks
CHAPTER 5 - Knights and Knaves
5.1 Logic Puzzles
5.2 Calculational Logic
5.2.1 Propositions
5.2.2 Knights andKnaves
5.2.3 Boolean Equality
5.2.4 HiddenTreasures
5.2.5 Equals forEquals
5.3 Equivalence and Continued Equalities
5.3.1 Examples oftheAssociativityofEquivalence
5.3.2 On Natural Language
5.4 Negation
5.4.1 Contraposition
5.4.2 Handshake Problems
5.4.3 Inequivalence
5.5 Summary
5.6 BibliographicRemarks
CHAPTER 6 - Induction
6.1 Example Problems
6.2 Cuttingthe Plane
6.3 Triominoes
6.4 LookingforPatterns
6.5 TheNeed forProof
6.6 FromVerification to Construction

CONTENTS
10.6 TheAlgorithm
10.7 Summary
10.8 Bibliographic Remarks
CHAPTER 11 - Knight's Circuit
11.1 Straight-Move Circuits
11.2 Supersquares
11.3 Partitioningthe Board
11.4 Summary
11.5 Bibliographic Remarks
PART II Mathematical Techniques
CHAPTER 12 - The Language of Mathematics
12.1 Variables, Expressions and Laws
12.2 Sets
12.2.1 The Membership Relation
12.2.2 The Empty Set
12.2.3 Types/Universes
12.2.4 Unionand Intersection
12.2.5 Set Comprehension
12.2.6 Bags
12.3 Functions
12.3.1 FunctionApplication
12.3.2 Binary Operators
12.3.3 OperatorPrecedence
12.4 TypesandType Checking
12.4.1 CartesianProductand DisjointSum
12.4.2 FunctionTypes
12.5 AlgebraicProperties
12.5.1 Symmetry
12.5.2 Zero andUnit
12.5.3 Idempotence
12.5.4 Associativity
12.5.5 Distributivity/Factorisation
12.5.6 Algebras
12.6 Boolean Operators
12.7 Binary Relations
12.7.1 Reflexivity
12.7.2 Symmetry
12.7.3 Converse
12.7.4 Transitivity
12.7.5 Anti-symmetry
12.7.6 Orderings

CONTENTS
15.4.1 IntegerDivision
15.4.2 Remainders and ModuloArithmetic
15.5 Exercises
CHAPTER 16 - Relations. Graphs and Path Algebras
16.1 Pathsin aDirected Graph
16.2 Graphs and Relations
16.2.1 Relation Composition
16.2.2 UnionofRelations
16.2.3 TransitiveClosure
16.2.4 ReflexiveTransitive Closure
16.3 Functional andTotalRelations
16.4 Path-FindingProblems
16.4.1 CountingPaths
16.4.2 Frequencies
16.4.3 ShortestDistances
16.4.4 AllPaths
16.4.5 Semirings and Operations onGraphs
16.5 Matrices
16.6 Closure Operators
16.7 Acyclic Graphs
16.7.1 Topological Ordering
16.8 Combinatorics
16.8.1 Basic Laws
16.8.2 CountingChoices
16.8.3 CountingPaths
16.9 Exercises
Solutions to Exercises
References
Index
Preface
In the modern, developed world our lifestyles, our livelihoods and even our lives
have become highly dependent oncomputer technology in a way thatwould have been
unimaginablefiftyyearsago.Underpinningthedevelopmentsthathavemadethispossible
hasbeenafundamentalchangeinthenatureofourproblem-solvingskills:inmanycases,
the solution to a difficult and challenging problem has to be formulated as a computer
program to be executed by a dumb machine. Not only have the size and complexity of
theproblemswetacklechangedbeyondmeasure,theprecisionandaccuracywithwhich
oursolutions need to be formulated have undergone a revolution. Algorithmic problem
solvingisabouttheskillsthatareneededto meetthe newchallengesthatwe face.
This book is based on a first-year, first-semester module that was first introduced at
the University of Nottingham in September 2003. Initially the module was optional
but, in 2006, itwas made compulsory for all first-year Computer Science and Software
Engineering students (as well as still being optional for students on other degree
programmes) and has remained so ever since. The module is taught at each of the
University’sthreecampuses in China,Malaysia and theUK.
The aim of the book is to instill good problem-solving skills, particularly but not
exclusively incaseswhere thesolutionto a probleminvolvesthedesignofanalgorithm.
Theapproachisproblem-driven:themainpartofthebookconsistsofaseriesofexamples
thatintroducetheprinciplesofalgorithmicproblemsolvingina systematicfashion.The
use of a problem-driven approach is crucial to exciting students’ natural propensity to
take on a challenge; however, examples without theory are meaningless, so the second
part of the book is about the mathematics that underpins the principles. Boxes within
thetextpointthereadertothemathematics thatis relevanttoa particularexample.
The presentation deviates from traditional mathematical practice in a number ofways.
The treatment of boolean algebra and logic, for example, is non-standard and some
notationisalso non-standard.These deviationsare, however, basedondevelopments in
algorithmic problem solving that have been well understood for at least twenty years.
(Althoughtwentyyearsisaveryshorttime-spaninthedevelopmentofmathematics,itisa
relativelylongperiodinthemoderncomputerage.)Potentialteacherswhoarenotalready
convinced orunfamiliarwith thematerialare askedto approach itwithanopen mind.
The majority of the book is class-tested but not all, and not all at the level ofthe first
year,firstsemesterofanundergraduatedegree. ForthreeyearsItaughtthemathematical
techniques (Part II ofthe book) in a module that was given in the same semester as the
PREFACE
moduleonalgorithmicproblemsolving(PartI),butthatisnolongerthecase.Also,both
parts of the book cover more material than I am able to cover in any one year. Some
topics (such as boolean algebra) could be presented at pre-university level but some (in
the laterchapters) are betterpostponed to lateryears.
Havingan excess ofexample problems has the advantage ofofferinga choice. My own
practice is to base assessment firmly on coursework that involves the students in active
problem solving and which I vary from year to year so that no two years are the same.
Model solutions to all exercises, which I have given to students for feedback purposes,
are included at the end ofthe text. Solutions are omitted, however, for some exercises
that I call “projects” and some projects that I set have not been included in the text.
(Model solutions to the projects aregiven to the students in hard-copy form in orderto
be abletoreuse themwithouttheriskofplagiarism.)
Whenteachingatopicsuchasthis,itisveryimportantthattheexamplesarechallenging
but within the grasp of the students. The problems I have chosen are most often ones
thatthereadershouldbeabletosolveinabrute-forcefashionwithoutanymathematical
knowledge but which can be solved much more effectively with the knowledge. The
hidden agenda in the book is to engage the readerin the process ofdoing mathematics:
theprocessofmodellingandcalculatingthesolutionstoproblemsevenwhenmathematics
does notseemto bedirectlyrelevant.
The presentation here is very strongly influenced by the writings of Edsger W. Dijk-
stra (1930—2002). Dijkstra is renowned for his contributions to algorithm design. In
addition, throughout his career he put much effort into trying to articulate and, by
so doing, improve mathematical method. My own view is that his contribution to
mathematical method is yet greater than his (phenomenal) contribution to computing
science. OccasionallyI haveusedhiswords (orparaphrasesofhiswords) withoutdirect
acknowledgement; like true proverbs, they deserve the acceptance thatonlycomes with
anonymity. One such is Dijkstra’s definition of mathematics as “the art of effective
reasoning” (see Chapter 12). I will never be able to emulate his problem-solving skills
but I do hope that by trying to continue his work on mathematical method, I might
encourage others tothinkmorecriticallyaboutanddeliberatelyarticulategoodandbad
problem-solvingskills.
I have received a lot of support from colleagues in the preparation of the book and
I would like to thank them all. I would particularly like to thank Diethard Michaelis
and David Gries, both ofwhom have given very detailed criticism and suggestions for
improvement. Particular thanks also go to those who have helped with the teaching
of the module: Siang Yew Chong and John Woodward have taught the module in
MalaysiaandChina,respectively,forseveralyears.JoaoFerreira alsotaughtthemodule
inNottinghamforoneyearwhenIwasonsabbaticalandhasassistedmeformanyyears
with classes and with marking coursework and examinations, as have Wei Chen and