数字建模 模拟一数独陈丽娟蒋鑫陈德进.docx

数字建模 模拟一数独陈丽娟蒋鑫陈德进.docx

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数字建模模拟一数独陈丽娟蒋鑫陈德进

CreatingSudokuPuzzles

Summary

Sudokupuzzleisanattractivegamewhichisverypopularintheworldnow.Atcurrent,creatingandevaluatingthedifficultylevelofSudokupuzzlesisasignificantproblem.InourpaperwedesignanalgorithmthatbasedonimprovementbackdatingmethodandcanproducedifferentdifficultySudokuwithauniquesolution,thenevaluatethecomplexityofit.

First,wedefinethebasicoperationoffindingthesmallestcandidatenumberandjudgingwhetherthelayoutisfinallayoutordeathlayout.Onthisbasis,usingthesearchalgorithmminimumcandidatenumberandbacktrackingexhaustivelystatespacetominimizealgorithmcomplexity.

Then,constructadecisiontreeandusebackdatingmethodtodefinedifficultyevaluationfromthreeaspects.Thethreeaspectsarethenumberofknowngridsinoriginallayout,thesmallestknownnumberofeachrowandeachline,thesearchnumberinsolvingthefinallayout.Throughalotofexperimentsweendowthemwithcorrespondingweights,andusetheweightstocalculatethedifficultygradeofeachoriginallayout.

Finally,designanalgorithmtoproduceanoriginallayoutwhichsatisfiestherulesofSudoku.Solveitandgetallthefinallayoutsofit,thenusebacktrackingtoseekanoriginallayoutwithuniquesolutionmeetingthechosendifficultylevel.

keywords:

thesearchalgorithmminimumcandidate;decisiontree;weights;backtracking

Content

1.Introduction

SudokuwasdevisedbymathematicianOlah,andnowitispopularathomeandabroad.Itisanattractivegamethatlotsofpeopleliketostudyitthoughbuyingbooks.Thereare93×3unitgridsintheoriginalSudoku,eachofwhichisaunit3×3grid.Eachgridcanhaveanumberof1to9togiveitasimpleconstraint,ormaybeempty.Therulesofthegameareasfollows[3]:

1.Eachrowofcellscontainstheintegers1throughto9exactlyonce;

2.Eachcolumnofcellscontainstheintegers1throughto9exactlyonce;

3.Each3×3squarecontainstheintegers1throughto9exactlyonce.

Inordertounderstandclearlywegiveasimpleexampleaspicture1shows:

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Figure1ThesimpleexampleofSudoku

2.DescriptionandAnalysis

WeshoulddevisealgorithmandmetricsofconstructingSudokupuzzleswhichcanbeextensibletoavaryingnumberofdifficultylevels,what’smoreensuretheuniquesolutionforeachSudoku.Themethodsofsolvingthegridsaregenerallyclassifiedintothevisual-fitmethodandcandidatenumbermethod.

Thevisual-fitmethodiscombiningthedigitalavailablewiththerows,columnsandpalacetheyarrayedin,andthenfindtheblanksthatcandecideauniquenumber.Thecandidatenumbermethodiscalculatingthenumberofcandidatesinsomeblank.Becausethevisual-fitmethodcanonlyagainstpartofthesimpleSudokusolution,inthepaperweusecandidatenumbermethod.

Ourprimaryaimismakingthesmallestalgorithmcomplexitywhiletherequirementsabovearesatisfied.Inordertoguaranteethealgorithmcomplexityminimum,weusethesmallestcandidatenumbermethod.What’smore,todescribelevelsofSudokupuzzles,wehavedifficultycriteriondefinition.

Wedescribethesolvingprocessasadecisiontreeasfigure2shows.Thenumbers（m,n）inthetreenodes,amongwhichmmeanstheblanksofcorrespondinglayoutandnmeansthenumberofthesmallestcandidates.Ifm>0andn=0,thatmeansthecorrespondinglayoutisadeathlayout;ifm=0andn=0,thatmeansthecorrespondinglayoutisafinallayout.ThisdecisiontreeindicatesthattheoriginallayoutS0hasS1andS2twosolutions,amongwhichfromS1.1toS1.5representthemiddlelayoutsofS1andfromS2.1toS2.5representthemiddlelayoutofS2,WhileS1.1isthesameasS2.1andS1.2isthesameasS2.2.

Figure2DecisiontreeofsolvingSudokuwith6spaces

3.Assumptions

1.Inthispaper,alllayoutsisa9×9grid,don'tinvolveothersizes.

2.Allstatisticsaboutrunningtimehavenothingtodowithcomputer,thatistosay,programs’runningtimeshouldbesameindifferentcomputers.

3.Therunningtimeisembodiedbythenumberofmiddlelayoutsbetweeneachoriginallayoutanditsallfinallayouts.

4.SymbolsandTerms

4.1Symbols

S:

thelayoutofSudoku;

i:

therownumberofunitgrid;

j:

thelinenumberofunitgrid;

m:

thenumberofblanksincorrespondinglayout;

n:

thenumberofthesmallestcandidates;

k:

thenumberoflayout.

4.2Terms[1]

OriginalLayout:

theunitgridstateatthebeginningofthegame,includeseveraldigitalandseveralspaces;

FinalLayout:

theunitgridstateattheendofthegame,includeseveraldigitalandnospacesexist;

MiddleLayout:

theunitgridstateintheprocessoffindingfinallayoutfromtheoriginallayout;

DeathLayout:

therearespacesinthelayout,butnomatterwhatdigitalfrom1to9fallinginthespacesallwillconflictwiththerulesofthegame;

Unitgrid:

thegridthatmakeupbya

gridwhichcontainthenumberfrom1to9onlyonce;

Candidates:

lotsofnumberthatcanbefilledintheunitgridthatmeansthenumbermustbedifferentfromthedigitalsinthesamerow,samelineandthesameunitgrid;

Solution:

anoriginallayoutwhichcanfinditsfinallayout

5.ModelDesignandSolve

5.1BasicOperation[1]

Obviously,thelayoutofSudokushouldbeexpressedbymatrixstructure.Todescribesubsequentalgorithmnowwedevisebasicoperationasfollows:

FunctionCandidates=Check（S,i,j）.

FunctionCheckusetocalculatethecandidatenumbervectoroftheunitrowilinejinSlayout.Candidates:

Function[i,j]=FindCandidate（S）.

FunctionFindCandidatesearchtheunit[i,j]whichhasthesmallestcandidatenumber.SupposetherearekspacesinlayoutSthatintheperformingoffunctionFindCandidate,thereneedtoinvokektimesfunctionCheck:

Functionflag=Success（S）.

FunctionSuccessisusedtoestimatewhetherSlayoutisfinallayout:

Functionflag=Failed（S）.

FunctionFailedisusedtoestimatewhethertheSlayoutisdeathlayout.

5.2Thesearchalgorithmbasedontheminimumnumberofcandidates

TheCandidatesEliminationTechniquesiswritingallthedigitalsthatmayappearintheblanksfirst,andthenthroughsomecommonlyusedalgorithmtocutcandidatenumberuntilgainthecertainuniquecandidatenumber.Itiswidelyusedincomputergeneratingpuzzleandproblemsolvinginpractice.Thisnotonlybecauseitrelativelyeasyprogramming,anditsalgorithmarealsogrowingwhichmakeitssolvingefficiencyandabilityraise.

InordertounderstandtheCandidatesEliminationmoreclearly,wegiveanexampleoffindingcandidatesinSudokuasfollow:

Figure3theexampleoffindingcandidates

Thebasicmethodtosolveoriginallayoutisusingbackdatingmethodtoillustratestatespaceexhaustively.Butwhichspaceshouldbefilledinwhichdigitalineachstepwillinfluencethestatespacegreatly,andtheninfluencetheefficiencyofthealgorithm.Generally,wethinkthestatespacetosolveoriginallayoutisa9forktreewhosedepthislessthan81[1].

Usingmoreforktreetraversalalgorithmframeandcombinethecalculationofthesmallestcandidatesnumber,wedevisealgorithmasfollows[1].

WriteoriginallayoutasS0,writethefirstKalayoutofthesolvingprocessasSk,andstorethedatastructureofmiddlelayoutasstackSS.TherecursivealgorithmofsolvingallthefinallayoutofSkdescribesasfollows:

1.If

thatistosaySkisthefinallayout,thenfromS1toSkwhichkeptinstackSSaresolvingprocessesfromoriginallayouttoanfinallayoutandreturntotheupperfunctioncalls;

2.If

thatistosaySkisastalemate,soreturntotheupperfunctioncalls;

3.Findthesmallestcandidatesunitgrid’spositioninSkthatis

;

4.Calculatethecandidatesvectorofposition[i,j]thatisCandidates=Check（Sk,i,j）;

Fillingeachcandidateintheemptygrids[i,j]ofSkinturntogainanewlayout

Sk+1,thenputSk+1instackSSandinvokeSolve（Sk+1）recursively,afterthatexecuteastackoperation.

Theflowchartoffunctionsolveisasfollow:

Figure4theprogramflowchartoffunctionSolve

5.3TheMeasurementforDifficultyGrades[2]

WeevaluatethedifficultyofaSudokufromthelogicinferenceofhumanandtheexhaustivesearchofcomputer.Duetotheplayer’sreasoningtotheunknowngridsbasedontheinformationprovidedbytheknowngrids,whiletheknowngrids’quantityanddistributiondecidedhowmuchinformationtheycanprovided.Thelessknowngrids,anditsdistributionismoreuneven,moreasymmetrythatthelogicinferenceoftheSudokushouldbemoredifficult.

ThustoaSudokutopic,wegivethreeevaluationprojects（aschart1）,evaluateitbymarkingitfrom0to4,andtogivethethreeprojectscorrespondingweights,thengetthesummationscoringofthetopic.Usetheintegerpartofthescoretoweighthegamers’difficultylevelneedasfollows:

Grade1:

thescorefrom0to1;

Grade2:

thescorefrom1to2;

Grade3:

thescorefrom2to3;

Grade4:

thescorefrom3to4.

Chart1evaluationofSudoku

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