数据结构与算法分析C版答案Word格式.docx

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14AnalysisTechniques88

15LimitstoComputation94

Preface

ContainedhereinarethesolutionstoallexercisesfromthetextbookAPractical

IntroductiontoDataStructuresandAlgorithmAnalysis,2ndedition.

FormostoftheproblemsrequiringanalgorithmIhavegivenactualcode.In

afewcasesIhavepresentedpseudocode.Pleasebeawarethatthecodepresented

inthismanualhasnotactuallybeencompiledandtested.WhileIbelievethealgorithms

tobeessentiallycorrect,theremaybeerrorsinsyntaxaswellassemantics.

Mostimportantly,thesesolutionsprovideaguidetotheinstructorastotheintended

answer,ratherthanusableprograms.

1

DataStructuresandAlgorithms

Instructor’snote:

Unliketheotherchapters,manyofthequestionsinthischapter

arenotreallysuitableforgradedwork.Thequestionsaremainlyintendedtoget

studentsthinkingaboutdatastructuresissues.

Thisquestiondoesnothaveaspecificrightanswer,providedthestudent

keepstothespiritofthequestion.Studentsmayhavetroublewiththeconcept

of“operations.”

Thisexerciseasksthestudenttoexpandontheirconceptofanintegerrepresentation.

AgoodanswerisdescribedbyProject,whereasingly-linked

listissuggested.Themoststraightforwardimplementationstoreseachdigit

initsownlistnode,withdigitsstoredinreverseorder.Additionandmultiplication

areimplementedbywhatamountstograde-schoolarithmetic.For

addition,simplymarchdowninparallelthroughthetwolistsrepresenting

theoperands,ateachdigitappendingtoanewlisttheappropriatepartial

sumandbringingforwardacarrybitasnecessary.Formultiplication,combine

theadditionfunctionwithanewfunctionthatmultipliesasingledigit

byaninteger.Exponentiationcanbedoneeitherbyrepeatedmultiplication

（notreallypractical）orbythetraditionalΘ（logn）-timealgorithmbasedon

thebinaryrepresentationoftheexponent.Discoveringthisfasteralgorithm

willbebeyondthereachofmoststudents,soshouldnotberequired.

AsampleADTforcharacterstringsmightlookasfollows（withthenormal

interpretationofthefunctionnamesassumed）.

Chap.1DataStructuresandAlgorithms

Some

InC++,thisis1

fors1<

s2;

0fors1=s2;

intstrcmp（Strings1,Strings2）

One’scomplimentstoresthebinaryrepresentationofpositivenumbers,and

storesthebinaryrepresentationofanegativenumberwiththebitsinverted.

Two’scomplimentisthesame,exceptthatanegativenumberhasitsbits

invertedandthenoneisadded（forreasonsofefficiencyinhardwareimplementation）.

ThisrepresentationisthephysicalimplementationofanADT

definedbythenormalarithmeticoperations,declarations,andothersupport

givenbytheprogramminglanguageforintegers.

AnADTfortwo-dimensionalarraysmightlookasfollows.

Matrixadd（MatrixM1,MatrixM2）;

Matrixmultiply（MatrixM1,MatrixM2）;

Matrixtranspose（MatrixM1）;

voidsetvalue（MatrixM1,introw,intcol,intval）;

intgetvalue（MatrixM1,introw,intcol）;

Listgetrow（MatrixM1,introw）;

OneimplementationforthesparsematrixisdescribedinSectionAnotherimplementation

isahashtablewhosesearchkeyisaconcatenationofthematrixcoordinates.

Everyproblemcertainlydoesnothaveanalgorithm.AsdiscussedinChapter15,

thereareanumberofreasonswhythismightbethecase.Someproblemsdon’t

haveasufficientlycleardefinition.Someproblems,suchasthehaltingproblem,

arenon-computable.Forsomeproblems,suchasonetypicallystudiedbyartificial

intelligenceresearchers,wesimplydon’tknowasolution.

Wemustassumethatby“algorithm”wemeansomethingcomposedofstepsare

ofanaturethattheycanbeperformedbyacomputer.Ifso,thananyalgorithm

canbeexpressedinC++.Inparticular,ifanalgorithmcanbeexpressedinany

othercomputerprogramminglanguage,thenitcanbeexpressedinC++,sinceall

（sufficientlygeneral）computerprogramminglanguagescomputethesamesetof

functions.

Theprimitiveoperationsare

（1）addingnewwordstothedictionaryand

（2）searching

thedictionaryforagivenword.Typically,dictionaryaccessinvolvessomesort

ofpre-processingofthewordtoarriveatthe“root”oftheword.

Atwentypagedocument（singlespaced）islikelytocontainabout20,000words.A

usermaybewillingtowaitafewsecondsbetweenindividual“hits”ofmis-spelled

words,orperhapsuptoaminuteforthewholedocumenttobeprocessed.This

meansthatacheckforanindividualwordcantakeabout10-20ms.Userswill

typicallyinsertindividualwordsintothedictionaryinteractively,sothisprocesscan

takeacoupleofseconds.Thus,searchmustbemuchmoreefficientthaninsertion.

Theusershouldbeabletofindacitybasedonavarietyofattributes（name,location,

perhapscharacteristicssuchaspopulationsize）.Theusershouldalsobeabletoinsert

anddeletecities.Thesearethefundamentaloperationsofanydatabasesystem:

search,insertionanddeletion.

Areasonabledatabasehasatimeconstraintthatwillsatisfythepatienceofatypical

user.Foraninsert,delete,orexactmatchquery,afewsecondsissatisfactory.Ifthe

databaseismeanttosupportrangequeriesandmassdeletions,theentireoperation

maybeallowedtotakelonger,perhapsontheorderofaminute.However,thetime

spenttoprocessindividualcitieswithintherangemustbeappropriatelyreduced.In

practice,thedatarepresentationwillneedtobesuchthatitaccommodatesefficient

processingtomeetthesetimeconstraints.Inparticular,itmaybenecessarytosupport

operationsthatprocessrangequeriesefficientlybyprocessingallcitiesinthe

rangeasabatch,ratherthanasaseriesofoperationsonindividualcities.

Studentsatthislevelarelikelyalreadyfamiliarwithbinarysearch.Thus,they

shouldtypicallyrespondwithsequentialsearchandbinarysearch.Binarysearch

shouldbedescribedasbettersinceittypicallyneedstomakefewercomparisons

（andthusislikelytobemuchfaster）.

TheanswertothisquestionisdiscussedinChapter8.Typicalmeasuresofcost

willbenumberofcomparisonsandnumberofswaps.Testsshouldincluderunning

timingsonsorted,reversesorted,andrandomlistsofvarioussizes.

Thefirstpartiseasywiththehint,butthesecondpartisratherdifficulttodowithout

astack.

a）boolcheckstring（stringS）{

intcount=0;

for（inti=0;

i<

length（S）;

i++）

if（S[i]==’（’）count++;

if（S[i]==’）’）{

if（count==0）returnFALSE;

count--;

}

if（count==0）returnTRUE;

elsereturnFALSE;

b）intcheckstring（StringStr）{

StackS;

if（S[i]==’（’）

（i）;

if（））returni;

（）;

if（））return-1;

elsereturn（）;

AnswerstothisquestionarediscussedinSection.

Thisissomewhatdifferentfromwritingsortingalgorithmsforacomputer,since

person’s“workingspace”istypicallylimited,asistheirabilitytophysicallymanipulate

thepiecesofpaper.Nonetheless,manyofthecommonsortingalgorithmshave

theiranalogstosolutionsforthisproblem.Mosttypicalanswerswillbeinsertion

sort,variationsonmergesort,andvariationsonbinsort.

AnswerstothisquestionarediscussedinChapter8.

2

MathematicalPreliminaries

（a）Notreflexiveifthesethasanymembers.Onecouldargueitissymmetric,

antisymmetric,andtransitive,sincenoelementviolateanyof

therules.

（b）

Notreflexive（foranyfemale）.Notsymmetric（considerabrotherand

sister）.Notantisymmetric（considertwobrothers）.Transitive（forany

3brothers）.

（c）

Notreflexive.Notsymmetric,andisantisymmetric.Nottransitive

（onlygoesonelevel）.

（d）

Notreflexive（fornearlyallnumbers）.Symmetricsincea

+b

=b

+a,

sonotantisymmetric.Transitive,butvacuouslyso（therecanbeno

distincta,b,andc

whereaRb

andbRc）.

（e）

Reflexive.Symmetric,sonotantisymmetric.Transitive（butsortof

vacuous）.

（f）

Reflexive–checkallthecases.Sinceitisonlytruewhenx

=y,it

istechnicallysymmetricandantisymmetric,butrathervacuous.Likewise,

itistechnicallytransitive,butvacuous.

Ingeneral,provethatsomethingisanequivalencerelationbyprovingthatit

isreflexive,symmetric,andtransitive.

（a）

Thisisanequivalencethateffectivelysplitstheintegersintooddand

evensets.Itisreflexive（x

+x

isevenforanyintegerx）,symmetric

（sincex

+y

=y

+x）andtransitive（sinceyouarealwaysaddingtwo

oddorevennumbersforanysatisfactorya,b,andc）.

Thisisnotanequivalence.Tobeginwith,itisnotreflexiveforany

integer.

Thisisanequivalencethatdividesthenon-zerorationalnumbersinto

positiveandnegative.Itisreflexivesincex

˙

x>

0.Itissymmetricsince

xy˙

=yx˙.Itistransitivesinceanytwomembersofthegivenclass

satisfytherelationship.

5

Chap.2MathematicalPreliminaries

Thisisnotanequivalancerelationsinc