AlgorithmsforSolvingRubiksCubes3Thuswesettlethediameterofthennnandnn1RubiksCubesuptoconstantfactorsTheseresultsaredescribedinSections4and3respectivelyn21puzzleAnotherpuzzlethatcanbedescrib ID: 487464
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2E.D.Demaine,M.L.Demaine,S.Eisenstat,A.Lubiw,A.WinslowartistknownasSpaceInvader.)Itisthebaneofmanycomputers,whichspentabout35CPUyearsdeterminingin2010thatthebestalgorithmtosolvetheworstcongurationrequiresexactly20moves|referredtoasGod'sNumber[22].Toamathematician,orastudenttakingabstractalgebra,theRubik'sCubeisashiningexampleofgrouptheory.ThecongurationsoftheRubik'sCube,orequivalentlythetransformationsfromonecongurationtoanother,formasubgroupofapermutationgroup,generatedbythebasictwistmoves.Thisperspectivemakesiteasiertoprove(andcompute)thatthecongurationspacefallsintotwoconnectedcomponents,accordingtotheparityofthepermutationonthecubies(theindividualsubcubesthatmakeupthepuzzle).See[7]forhowtocomputethenumberofelementsinthegroupgeneratedbythebasicRubik'sCubemoves(oranysetofpermutations)inpolynomialtime.Toatheoreticalcomputerscientist,theRubik'sCubeanditsmanygen-eralizationssuggestseveralnaturalopenproblems.WhataregoodalgorithmsforsolvingagivenRubik'sCubepuzzle?Whatisanoptimalworst-caseboundonthenumberofmoves?Whatisthecomplexityofoptimizingthenumberofmovesrequiredforagivenstartingconguration?AlthoughGod'sNumberisknowntobe20forthe333,theoptimalsolutionofeachcongurationinthisconstant-sizepuzzlestillhasnotbeencomputed[22].Whilecomputingtheexactbehaviorforlargercubesisoutofthequestion,howdoestheworst-casenumberofmovesandcomplexityscalewiththesidelengthsofthecube?Inparallelwithourwork,thesequestionswererecentlyposedbyAndyDruckerandJe Erickson[4].Scalabilityisimportantgiventhecommerciallyavailable444Rubik'sRevenge[25];555Professor'sCube[13];the666and777V-CUBEs[27];LeslieLe's3D-printed121212[14];andOskarvanDeventer's171717OvertheTopandhis2220OverlapCube,bothavailablefrom3Dprintershapeways[28].Diameter/God'sNumber.ThediameterofthecongurationspaceofaRubik'sCubeseemsdiculttocaptureusingjustgrouptheory.Ingeneral,asetofpermutations(moves)cangenerateagroupwithsuperpolynomialdiameter[3].Ifwerestricteachgenerator(move)tomanipulateonlykelements,thenthediameterisO(nk)[16],butthisgivesveryweak(superexponential)upperboundsfornnnandnn1Rubik'sCubes.Fortunately,weconrmthatthegeneralapproachtakenbyfolkalgorithmsforsolvingRubik'sCubesofvariousxedsizescanbegeneralizedtoperformaconstantnumberofmovespercubie,foranupperboundofO(n2).Thisresultisessentiallystandard,butwetakecaretoensurethatallcasescanbehandled.Surprisingly,thisboundisnotoptimal.Eachtwistmoveinthennnandnn1Rubik'sCubessimultaneouslytransformsn(1)cubies(withtheexpo-nentdependingonthedimensionsandwhetheramovetransformsaplaneorahalf-space).Thispropertyo ersaformofparallelismforsolvingmultiplecubiesatonce,totheextentthatmultiplecubieswantthesamemovetobeappliedataparticulartime.Weshowthatthisparallelismcanbeexploitedtoreducethenumberofmovesbyalogarithmicfactor,toO(n2=logn).Furthermore,aneasycountingargumentshowsanaverage-caselowerboundof (n2=logn). AlgorithmsforSolvingRubik'sCubes3Thuswesettlethediameterofthennnandnn1Rubik'sCubes,uptoconstantfactors.TheseresultsaredescribedinSections4and3,respectively.n2�1puzzle.AnotherpuzzlethatcanbedescribedasapermutationgroupgivenbygeneratorscorrespondingtovalidmovesisthenngeneralizationoftheclassicFifteenPuzzle.Thisn2�1puzzlealsohaspolynomialdiameter,thoughwithoutanyformofparallelism,thediameterissimply(n3)[20].Interestingly,computingtheshortestsolutionfromagivencongurationofthepuzzleisNP-hard[21].Moregenerally,givenasetofgeneratorpermutations,itisPSPACE-completetondtheshortestsequenceofgeneratorswhoseproductisagiventargetpermutation[5,11].ThesepapersmentiontheRubik'sCubeasmotivation,butneitheraddressesthenaturalquestion:isitNP-hardtosolveagivennnnornn1Rubik'sCubeusingthefewestpossiblemoves?Althoughthennnproblemwasposedasearlyas1984[2,21],bothquestionsremainopen[12].Wegivepartialprogresstowardhardness,aswellasapolynomial-timeexactalgorithmforaparticulargeneralizationoftheRubik'sCube.Optimizationalgorithms.WegiveonepositiveandonenegativeresultaboutndingtheshortestsolutionfromagivencongurationofageneralizedRubik'sCubepuzzle.Onthepositiveside,weshowinSection6howtocomputetheex-actoptimumfornO(1)O(1)Rubik'sCubes.Essentially,weprovestructuralresultsabouthowanoptimalsolutiondecomposesintomovesinthelongdimen-sionandthetwoshortdimensions,andusethisstructuretoobtainadynamicprogram.ThisresultmayproveusefulforoptimallysolvingcongurationsofOskarvanDeventer's2220OverlapCube[28],butitdoesnotapplytothe333Rubik'sCubebecauseweneedntobedistinctfromtheothertwosidelengths.Onthenegativeside,weproveinSection5thatitisNP-hardtondanoptimalsolutiontoasubsetofcubiesinannn1Rubik'sCube.Phraseddi erently,optimallysolvingagivennn1Rubik'sCubecongurationisNP-hardwhenthecolorsandpositionsofsomecubiesareignored(i.e.,theyarenotconsideredindeterminingwhetherthecubeissolved).2CommonDenitionsWebeginwithsometerminology.An`mnRubik'sCubeiscomposedof`mncubies,eachofwhichhassomeposition(x;y;z),wherex2f0;1;:::;`�1g,y2f0;1;:::;m�1g,andz2f0;1;:::;n�1g.Eachcubiealsohasanorientation.EachcubieinaRubik'sCubehasacoloroneachvisibleface.Therearesixcolorsintotal.WesaythataRubik'sCubeissolvedwheneachfaceofthecubeisthesamecolor,uniqueforeachface.Anedgecubieisanycubiewhichhasatleasttwovisiblefaceswhichpointinperpendiculardirections.Acornercubieisanycubiewhichhasatleastthreevisiblefaceswhichallpointinperpendiculardirections.AsliceofaRubik'sCubeisasetofcubiesthatmatchinonecoordinate(e.g.allofthecubiessuchthaty=1).AlegalmoveonaRubik'sCubeinvolves 10E.D.Demaine,M.L.Demaine,S.Eisenstat,A.Lubiw,A.Winslow x1x2x3~x1~x2y1y2y3~y1~y2~y3Fig.2.AsampleofthebetweennessgadgetfromLemma1.Importantcubiesareorange(solved)andblue(unsolved).Unimportantcubiesarewhite.AnyidealsolutionmusteitherhaveI1(x1)I1(x2)I1(x3)orI1(x3)I1(x2)I1(x1).Forourhardnessreduction,wedevelopagadget(depictedinFig.2)whichforcesabetweennessconstraintontheorderingofthreedi erentrowmoves:Lemma1.Giventhreecolumnsx1;x2;x3bn=2c,thereisagadgetusingsixextrarowsandtwoextracolumnsensuringthatI1(x2)liesbetweenI1(x1)andI1(x3).ThisgadgetalsoforcesI2(x2)I2(x1),I2(x2)I2(x3),andmaxx2fx1;x2;x3gI1(x)minx2fx1;x2;x3gI2(x):ThebetweennessproblemisaknownNP-hardproblem[8,19].Inthisproblem,wearegivenasetoftriples(a;b;c),andwishtondanorderingonallitemssuchthat,foreachtriple,eitherabcorcba.Inotherwords,foreachtriple,bshouldliebetweenaandcintheoverallordering.Lemma1givesusagadgetwhichwouldatrstseemtobeperfectlysuitedtoareductionfromthebetweennessproblem.However,becausethelemmaplacesadditionalrestrictionsontheorderofallmoves,wecannotreducedirectlyfrombetweenness.Instead,weprovideareductionfromanotherknownNP-hardproblem,Not-All-Equal3-SAT[8,23].Inthisproblem,sometimesknownas6=-SAT,theinputisa3-CNFformula andthegoalistodeterminewhetherthereexistsanassignmenttothevariablesof suchthatthereisatleastonetrueliteralandonefalseliteralineveryclause.Ourreductionfrom6=-SATtoidealRubiksolutionscloselyfollowsthereductionfromhypergraph2-coloringtobetweenness[19]. AlgorithmsforSolvingRubik'sCubes11Theorem5.Givena6=-SATinstance ,itispossibletocomputeintimepoly-nomialinthesizeof annn1congurationandasubsetofthecubiesthathasanidealsolutionifandonlyif hasasolution,i.e.,belongsto6=-SAT.6OptimallySolvinganO(1)O(1)nRubik'sCubeForthec1c2nRubik'sCubewithc16=n6=c2,theasymmetryofthepuzzleleadstoafewadditionaldenitions.Wecallasliceshortifthematchingcoordinateisz;otherwise,asliceislong.Ashortmoveinvolvesrotatingashortslice;alongmoveinvolvesrotatingalongslice.Wedenecubieclusteritobethepairofslicesz=iandz=(n�1)�i.Thisdenitionmeansthatanyshortmovea ectsthepositionandorientationofcubiesinexactlyonecubiecluster.Eachshortmovea ectsexactlyonecluster.Hence,intheoptimalsolution,thenumberofshortmovesa ectingaparticularclusterisatmostthenumberofcongurationsofthatcluster.EachclusterhasO(1)congurations,soinanyoptimalsolution,anyparticularshortmovewillbeperformedO(1)times.Anysequenceoflongmovescorrespondstoanarrangementofc1c2blocksofcubieswithdimensions11n.Wecalleachsucharrangementalongcongu-ration.Thereareaconstantnumberoflongcongurations,sotheremustexistalongmovetour:aconstant-lengthsequenceoflongmoveswhichpassesthrougheverylongcongurationbeforereturningtotheinitiallongconguration.Thee ectofashortmovedependsonlyonthecurrentlongconguration.Hence,tosolveaparticularc1c2npuzzle,itissucienttoknowasequenceofshortmovesforeachcluster,annotatedwiththelongcongurationthateachsuchmoveshouldbeperformedin.Ifwehavesuchasequenceforeachcluster,wemayconstructafullsolutiontothepuzzlebyrepeatedlyperformingalongmovetour,andinsertingshortmovesintotheappropriateplaces.Thenweareguaranteedtobeabletoperformthekthshortmoveforeveryclusterduringthekthlongmovetour.Hence,thenumberoflongmovetoursnecessaryisboundedbythemaximumlengthofanyshortmovesequence,whichisO(1)inanyoptimalsolution.Therefore,anyoptimalsolutioncontainsO(1)longmoves.Thisboundonthenumberoflongmovesallowsustoconstructanalgorithmthatdoesthefollowing:Theorem6.Givenanyc1c2nRubik'sCubeconguration,itispossibletondtheoptimalsolutionintimepolynomialinn.References1.WorldCubeAssociation.Ocialresults.http://www.worldcubeassociation.org/results/,2010.2.StephenA.Cook.Cancomputersroutinelydiscovermathematicalproofs?Pro-ceedingsoftheAmericanPhilosophicalSociety,128(1):40{43,1984.3.JamesR.DriscollandMerrickL.Furst.Onthediameterofpermutationgroups.InProceedingsofthe15thAnnualACMSymposiumonTheoryofcomputing,pages152{160,1983. 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