4 of Algorithm Design by Kleinberg Tardos brPage 2br Finding closest pair of points Problem Given a set of points 64257nd the pair of points that are closest together brPage 3br Goal Brute force gives an algorithm just check ever pair of points Ca ID: 51364
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Findingclosestpairofpoints Problem Givenasetofpointsfp1;:::;pngndthepairofpointsfpi;pjgthatareclosesttogether. GoalBruteforcegivesanO(n2)algorithm:justcheckeverpairofpoints.Canwedoitfaster?Seemslikeno:don'twehavetocheckeverypair?Infact,wecanndtheclosestpairinO(nlogn)time.What'sareasonablerststep? Merge:thehardcaseLetd=minfdleft;drightg. dwouldbetheanswer,exceptmaybeLsplitaclosepair! SlabMightContainAllPointsLetSybeanarrayofthepointsinthatregion,sortedbydecreasingy-coordinatevalue.Symightcontainallthepoints,sowecan'tjustcheckeverypairinsideit. Theorem SupposeSy=p1;:::;pm.Ifdist(pi;pj)dthenj�i15. Inotherwords,iftwopointsinSyarecloseenoughintheplane,theyarecloseinthearraySy. Proof,1Dividetheregionupintosquareswithsidesoflengthd=2: Howmanypointsineachbox? Atmost1becauseeachboxiscompletelycontainedinonehalfandnotwopointsinahalfarecloserthand. LinearTimeMergeTherefore,wecanscanSyforpairsofpointsseparatedbydinlineartime.ClosestPair(Px,Py):if|Px|==2:returndist(Px[1],Px[2])//based1=ClosestPair(FirstHalf(Px,Py))//divided2=ClosestPair(SecondHalf(Px,Py))d=min(d1,d2)Sy=pointsinPywithindofL//mergeFori=1,...,|Sy|:Forj=1,...,15:d=min(dist(Sy[i],Sy[j]),d)Returnd