Here are represen ting this as 2D problem but in general it can an dimensional problem oin ts are represen ted their co ordinates or this problem co ordinates eing in tegral on mak uc of di57355erence though will assume that ha real co ordinates 11 ID: 67872
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ComputationalGeometry,Lectures3,4ClosestPairProblemScribeVarunNayyar1ClosestPairProblemGivenasetofnpointsdeterminepointsa,bsuchthattheinterpointdistance(Euclidean)istheminimumamongallpossiblepairofpoints.Herewearerepresentingthisasa2-Dproblembutingeneralitcanbeanndimensionalproblem.Pointsarerepresentedbytheircoordinates.Forthisproblemcoordinatesbeingintegralwontmakemuchofadi erence,thoughwewillassumethatwehaverealcoordinates.1.1InterPointDistanceInterpointdistancebetweentwopointsPi(xi;yi)andPj(xj;yj)isgivenasfollows:q(xi xj)2+(yi yj)2.Thiscaneasilybeextendedtoanndimensionalcase.1.2TrivialMethodForanydimensiond,wecansolvethisbycomputingtheminimumofn2pairs.Thisgivesarunningtimeof(n2).Asweincreasethedimensionthecomputationformindistance1 scalesup,hencetherunningtimeof(n2d).1.3ProbleminOneDimensionInonedimensionallthepointsareinalineafterwesortthemin(nlogn).Aftersortingwecangothroughthemto ndtheconsecutivepointswithleastseparation.Thustheprobleminonedimensionissolvedin(nlogn)timewhichismainlydominatedbysortingtime.2DivideAndConquerConsiderthe2-Dproblem.Wecan ndthemedianline(x-coordinate)in(n)time.Nowlet1betheminimuminterpointdistanceinthelefthalf.Similarlylet2betheminimuminterpointdistanceintherighthalf.Wehave=min(1;2).Thepairswherethelefthalfisontheleftsideandtherighthalfontherightsideformaseparatecase.Weneednotworryaboutallsuchpoints,onlypointslyingwiththe2 stripareofconcerntous.Allotherswillhavehaveinterpointdistancegreaterthenandcanbeeliminatedfromconsideration.Thekindofrecurrencewearelookingwillbeofthefollowingtype:T(n)=2T(n2)+(n)+MiddleCasewhere(n)timefor ndingthemedianandtheMiddleCasereferstothenumberofcom-parisonsneededtobedonewithinthe2 Strip.2 2.1NumberofComparisonswithintheDeltaStripSparsityCondition:Wecannothavetwomanypointswithinonesideofthe2 StripasitwillmaketheinterpointdistanceonthathalflessthanStripitself,whichisacontradictionaccordingtothede nitionofStrip.Whatisthemaximumnumberofpointsinthe2 StripXsquarewheretheinterpointDistanceis?2.2SolvingTechniquefor2DProjectallpointontheverticalline.ForeveryprojectedpointPweneedto ndouthowmanyotherpointsontheverticallinewemustcompareitwithasshownintheillustrationbelow.ThegreenpointsaretheProjectionofthepointsontherighthalfoftheStripwhiletheredpointsareForthelefthalf.Foreverypointweneedtogoupanddownandcomparepointslyingwithinthatintervalonly.ThusforeverypointwehavetodoaconstantCnumberofcomparisons.WecansortthemonthebasisofycoordinatebeforedoingtheDivideandConquerprocedure.Thustheoveralltimewillbe(nlogn)+T(n)...whereT(n)=2T(n=2)+0(n)+Cd(n).ThisconstantCdscalesupwithdimension'd'.3 2.33andHigherDimensionsForthesameproblemin3-Dwecanprojectpointsonbothhalfwithindeltadistanceonaplane.Thusweseebyprojectingwearereducingtheproblembyonedimension.AgainbecauseofsparsityconditiontherearenomorethanaconstantnumberofpointssayCd(dependenton'd')ina2 X2 Square.2.4Lemma:Inthed-dimensionitispossibleto ndapartitioningplaneHwiththefollowingproperties:Thesizeofthesmallersetn2dandlargestset(1 12d)nIfthemininterpointdistanceisthenthe2slicedoesnotcontainmorethanCdn(1 1d)points.(Noteisunknown).Wecanapplyasecondrecursiontotheinnerplacethussolvingthesameproblemin2-Dnow.Therecursiveequationwouldlooksomethinglikethisinthatcase:T(n)2T(n=2)+M(k)+O(n)whereM(k)isthetimetomergekpointsinthe2 Slab.Here,M(k)equalsO(klogk)Ifweusetheabovelemmachoosethespeciallinetopartitionk=O(n23).SoM(n23)isO(n).2.5AlternateMethodThismethodisforthe2Dsub-problemreducedfromprojecting3Dproblemontoaplane.Theproceduremaintainaseparatedgridasshownbelow:ForeverypointwehavetolookattheEightAdjacentcellsonly.Sparcityconditionguaranteesnottoomanypointswithinacell.Soforeachpointweneedaconstanttimetosearchpointswithineightadjacentcells.Hencefornpointswewouldneed(n).4 5