计算机算法分析与设计课程综合实验.docx

计算机算法分析与设计课程综合实验.docx

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计算机算法分析与设计课程综合实验

算法分析与设计课程综合实验

DesignandAnalysisofAlgorithms

1MapRouting

要求：

Mandatory.

实验目的：

ImplementtheclassicDijkstra'sshortestpathalgorithmandoptimizeitformaps.Suchalgorithmsarewidelyusedingeographicinformationsystems（GIS）includingMapQuestandGPS-basedcarnavigationsystems.

实验内容及要求：

Maps.Forthisassignmentwewillbeworkingwithmaps,orgraphswhoseverticesarepointsintheplaneandareconnectedbyedgeswhoseweightsareEuclideandistances.Thinkoftheverticesascitiesandtheedgesasroadsconnectedtothem.Torepresentamapinafile,welistthenumberofverticesandedges,thenlistthevertices（indexfollowedbyitsxandycoordinates）,thenlisttheedges（pairsofvertices）,andfinallythesourceandsinkvertices.Forexample,Input6representsthemapbelow:

Dijkstra'salgorithm. Dijkstra'salgorithmisaclassicsolutiontotheshortestpathproblem.Itisdescribedinsection24.3inCLRS.Thebasicideaisnotdifficulttounderstand.Wemaintain,foreveryvertexinthegraph,thelengthoftheshortestknownpathfromthesourcetothatvertex,andwemaintaintheselengthsinapriorityqueue.Initially,weputalltheverticesonthequeuewithanartificiallyhighpriorityandthenassignpriority0.0tothesource.Thealgorithmproceedsbytakingthelowest-priorityvertexoffthePQ,thencheckingalltheverticesthatcanbereachedfromthatvertexbyoneedgetoseewhetherthatedgegivesashorterpathtothevertexfromthesourcethantheshortestpreviously-knownpath.Ifso,itlowerstheprioritytoreflectthisnewinformation.

Hereisastep-by-stepdescriptionthatshowshowDijkstra'salgorithmfindstheshortestpath0-1-2-5from0to5intheexampleabove.

process0（0.0）

lower3to3841.9

lower1to1897.4

process1（1897.4）

lower4to3776.2

lower2to2537.7

process2（2537.7）

lower5to6274.0

process4（3776.2）

process3（3841.9）

process5（6274.0）

Thismethodcomputesthelengthoftheshortestpath.Tokeeptrackofthepath,wealsomaintainforeachvertex,itspredecessorontheshortestpathfromthesourcetothatvertex.ThefilesEuclideanGraph.java,Point.java,IndexPQ.java,IntIterator.java,andDijkstra.javaprovideabarebonesimplementationofDijkstra'salgorithmformaps,andyoushouldusethisasastartingpoint.TheclientprogramShortestPath.javasolvesasingleshortestpathproblemandplotstheresultsusingturtlegraphics.TheclientprogramPaths.javasolvesmanyshortestpathproblemsandprintstheshortestpathstostandardoutput.TheclientprogramDistances.javasolvesmanyshortestpathproblemsandprintsonlythedistancestostandardoutput.

Yourgoal. OptimizeDijkstra'salgorithmsothatitcanprocessthousandsofshortestpathqueriesforagivenmap.Onceyoureadin（andoptionallypreprocess）themap,yourprogramshouldsolveshortestpathproblemsinsublineartime.Onemethodwouldbetoprecomputetheshortestpathforallpairsofvertices;howeveryoucannotaffordthequadraticspacerequiredtostoreallofthisinformation.Yourgoalistoreducetheamountofworkinvolvedpershortestpathcomputation,withoutusingexcessivespace.Wesuggestanumberofpotentialideasbelowwhichyoumaychoosetoimplement.Oryoucandevelopandimplementyourownideas.

Idea1. ThenaiveimplementationofDijkstra'salgorithmexaminesallVverticesinthegraph.Anobviousstrategytoreducethenumberofverticesexaminedistostopthesearchassoonasyoudiscovertheshortestpathtothedestination.Withthisapproach,youcanmaketherunningtimepershortestpathqueryproportionaltoE'logV'whereE'andV'arethenumberofedgesandverticesexaminedbyDijkstra'salgorithm.However,thisrequiressomecarebecausejustre-initializingallofthedistancesto∞wouldtaketimeproportionaltoV.Sinceyouaredoingrepeatedqueries,youcanspeedthingsupdramaticallybyonlyre-initializingthosevaluesthatchangedinthepreviousquery.

Idea2. YoucancutdownonthesearchtimefurtherbyexploitingtheEuclideangeometryoftheproblem,asdescribedinsection21.5inthebookofAlgorithminCPartV.Forgeneralgraphs,Dijkstra'srelaxesedgev-wbyupdatingd[w]tothesumofd[v]plusthedistancefromvtow.Formaps,weinsteadupdated[w]tobethesumofd[v]plusthedistancefromvtowplustheEuclideandistancefromwtodminustheEuclideandistancefromvtod.ThisisknownastheA*algorithm.Thisheuristicsaffectsperformance,butnotcorrectness.

Idea3. Useafasterpriorityqueue.Thereissomeroomforoptimizationinthesuppliedpriorityqueue.YoucouldalsoconsiderusingamultiwayheapasinSedgewickProgram20.10.

Testing. Thefileusa.txtcontains87,575intersectionsand121,961roadsinthecontinentalUnitedStates.Thegraphisverysparse-theaveragedegreeis2.8.Yourmaingoalshouldbetoanswershortestpathqueriesquicklyforpairsofverticesonthisnetwork.Youralgorithmwilllikelyperformdifferentlydependingonwhetherthetwoverticesarenearbyorfarapart.Weprovideinputfilesthattestbothcases.Youmayassumethatallofthexandycoordinatesareintegersbetween0and10,000.

实验类型：

Verification.

适用对象：

UndergraduateforComputerSchool

2DocumentDistanceProblem

要求：

Mandatory.

实验目的：

Designandimplementthedocumentdistanceproblemandoptimizeitfordata.

实验内容及要求：

LetDbeatextdocument（e.g.thecompleteworksofWilliamShakespeare）.

Awordisaconsecutivesequenceofalphanumericcharacters,suchas"Hamlet"or"2007".We'lltreatallupper-caselettersasiftheyarelower-case,sothat"Hamlet"and"hamlet"arethesameword.Wordsendatanon-alphanumericcharacter,so"can't"containstwowords:

"can"and"t".

ThewordfrequencydistributionofadocumentDisamappingfromwordswtotheirfrequencycount,whichwe'lldenoteasD（w）.

WecanviewthefrequencydistributionDasvector,withonecomponentperpossibleword.Eachcomponentwillbeanon-negativeinteger（possiblyzero）.

Thenormofthisvectorisdefinedintheusualway:

.

Theinner-productbetweentwovectorsDandD'isdefinedasusual.

.

Finally,theanglebetweentwovectorsDandD'isdefined:

Thisangle（inradians）willbeanumberbetween0and

sincethevectorsarenon-negativeineachcomponent.Clearly,

angle（D,D）=0.0

forallvectorsD,and

angle（D,D'）=π/2

ifDandD'havenowordsincommon.

Example:

Theanglebetweenthedocuments"Tobeornottobe"and"Doubttruthtobealiar"is

Wedefinethedistancebetweentwodocumentstobetheanglebetweentheirwordfrequencyvectors.

Thedocumentdistanceproblemisthustheproblemofcomputingthedistancebetweentwogiventextdocuments.

Aninstanceofthedocumentdistanceproblemisthepairofinputtextdocuments.

3EditDistance

要求：

Mandatory.

实验目的：

Manywordprocessorsandkeywordsearchengineshaveaspellingcorrectionfeature.Ifyoutypeinamisspelledwordx,thewordprocessororsearchenginecansuggestacorrectiony.Thecorrectionyshouldbeawordthatisclosetox.Onewaytomeasurethesimilarityinspellingbetweentwotextstringsisby“editdistance”.Thenotionofeditdistanceisusefulinotherfieldsaswell.Forexample,biologistsuseeditdistancetocharacterizethesimilarityofDNAorproteinsequences.

实验内容及要求：

Theeditdistanced（x,y）oftwostringsoftext,x[1..m]andy[1..n],isdefinedtobetheminimumpossiblecostofasequenceof“transformationoperations”（definedbelow）thattransformsstringx[1..m]intostringy[1..n].Todefinetheeffectofthetransformationoperations,weuseanauxiliarystringz[1..s]thatholdstheintermediateresults.Atthebeginningofthetransformationsequences=mandz[1..s]=x[1..m]（i.e.,westartwithstringx[1..m]）.Attheendofthetransformationsequence,weshouldhaves=nandz[1..s]=y[1..n]（i.e.,ourgoalistotransformintostringy[1..n]）.Throughoutthetransformation,wemaintainthecurrentlengthsofstringz,aswellasacursorpositioni,i.e.,anindexintostringz.Theinvariant1is+1holdsatalltimesduringthetransformation.（Noticethatthecursorcanmoveonespacebeyondtheendofthestringzinordertoallowinsertionattheendofthestring.）

Eachtransformationoperationmayalterthestringz,thesizes,andthecursorpositioni.Eachtransformationoperationalsohasanassociatedcost.Thecostofasequenceoftransformationoperationsisthesumofthecostsoftheindividualoperationsonthesequence.Thegoaloftheedit-distanceproblemistofindasequenceoftransformationoperationofminimumcostthattransformsx[1..m]intoy[1..n].

Therearefivetransformationoperations:

OperationCostEffect

left0Ifi=1thendonothing.Otherwise,setii-1

right0Ifi=s+1thendonothing.Otherwise,setii-1.

replace4Ifi=s+1thendonothing.Otherwise,replacethecharacterunderthecursorbyanothercharactercbysettingz[i]c,andthenincrementingi.

delete2Ifi=s+1thendonothing.Otherwise,deletethecharactercunderthecursorbysettingz[i..s]z[i+1..s+1]anddecrementings.Thecursorpositionidoesnotchange.

insert3Insertthecharactercintostringzbyincrementings,settingz[i+1..s]z[i..s-1],settingz[i]c,andthenincrementingindexi.

Asanexample,onewaytotransformthesourcestringalgorithmtothetargetstringanalysisistousethesequenceofoperationsshowninTable1,wherethepositionoftheunderlinedcharacterrepresentsthecursorpositioni.ManyothersequencesoftransformationoperationsalsotransformalgorithmtoanalysisthesolutioninTable1isnotuniqueandsomeothersolutions