Heap orderingLeftcomplete Binary Tree143169101 Figure11HeapNexttimewewillshowhowthepriorityqueueoperationsareimplementedforaheapLecture13HeapSortTuesdayMar101998Chapt7inCLRRecallthataheapisa ID: 478226
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LectureNotesCMSC251Aheapisrepresentedasanbinarytree.Thismeansthatallthelevelsofthetreearefullexceptthebottommostlevel,whichislledfromlefttoright.Anexampleisshownbelow.Thekeysofaheaparestoredinsomethingcalledheaporder.Thismeansthatforeachnode,otherthantheroot,Parent.Thisimpliesthatasyoufollowanypathfromaleaftotherootthekeys Heap orderingLeft-complete Binary Tree143169101 Figure11:Heap.Nexttimewewillshowhowthepriorityqueueoperationsareimplementedforaheap.Lecture13:HeapSort(Tuesday,Mar10,1998)Chapt7inCLR.Recallthataheapisadatastructurethatsupportsthemainpriorityqueueoperations(insertandextractmax)in LectureNotesCMSC251 1 5 6 7 8 9 161410 8 7 9 4 2 1 Figure12:Storingaheapinanarray.Weclaimthattoaccesselementsoftheheapinvolvessimplearithmeticoperationsonthearrayindices.Inparticularitiseasytoseethefollowing.LeftRightParentIsLeafLeft.(Thatis,if'sleftchildisnotinthetree.)IsRoot==1Forexample,theheaporderingpropertycanbestatedas“forall,if(notIsRoot)thenthenParententi]”.Soisaheapabinarytreeoranarray?Theansweristhatfromaconceptualstandpoint,itisabinarytree.However,itisimplemented(typically)asanarrayforspaceefciency.MaintainingtheHeapProperty:Thereisoneprincipaloperationformaintainingtheheapproperty.Itis.(Inotherbooksitissometimescalledsiftingdown.)Theideaisthatwearegivenanelementoftheheapwhichwesuspectmaynotbeinvalidheaporder,butweassumethatallofothertheelementsinthesubtreerootedatthiselementareinheaporder.Inparticularthisrootelementmaybetoosmall.Toxthiswe“sift”itdownthetreebyswappingitwithoneofitschildren.Whichchild?Weshouldtakethelargerofthetwochildrentosatisfytheheaporderingproperty.Thiscontinuesrecursivelyuntiltheelementiseitherlargerthanbothitschildrenoruntilitsfallsallthewaytotheleaflevel.Hereisthepseudocode.Itisgiventheheapinthearray,andtheindexofthesuspectedelement,andthecurrentactivesizeoftheheap.Theelementelementmax]issettothemaximumofofi]andittwochildren.Ifthenweswapapi]andA[max]andthenrecurseon Heapify(arrayA,inti,intm){//siftdownA[i]inA[1..m]l=Left(i)//leftchildr=Right(i)//rightchildmax=iif(lmandA[l]&#x=-60;�A[max])max=l//leftchildexistsandlargerif(rmandA[r]&#x=-60;�A[max])max=r//rightchildexistsandlargerif(max!=i){//ifeitherchildlargerswapA[i]withA[max]//swapwithlargerchildHeapify(A,max,m)//andrecurse LectureNotesCMSC251 SeeFigure7.2onpage143ofCLRforanexampleofhowHeapifyworks(inthecasewhere=10Weshowtheexecutiononatree,ratherthanonthearrayrepresentation,sincethisisthemostnaturalwaytoconceptualizetheheap.Youmighttrysimulatingthissamealgorithmonthearray,toseehowitworksatanerdetails.NotethattherecursiveimplementationofHeapifyisnotthemostefcient.Wehavedonesobecausemanyalgorithmsontreesaremostnaturallyimplementedusingrecursion,soitisnicetopracticethishere.Itispossibletowritetheprocedureiteratively.Thisisleftasanexercise.TheHeapSortalgorithmwillconsistoftwomajorparts.Firstbuildingaheap,andthenextractingthemaximumelementsfromtheheap,onebyone.WewillseehowtouseHeapifytohelpusdobothofHowlongdoesHepifytaketorun?ObservethatweperformaconstantamountofworkateachlevelofthetreeuntilwemakeacalltoHeapifyatthenextlowerlevelofthetree.Thuswedoworkforeachlevelofthetreewhichwevisit.Sincetherearelevelsaltogetherinthetree,thetotaltimeforHeapifyis.(Itisnotsince,forexample,ifwecallHeapifyonaleaf,thenitwillterminateinBuildingaHeap:WecanuseHeapifytobuildaheapasfollows.Firstwestartwithaheapinwhichtheelementsarenotinheaporder.Theyarejustinthesameorderthattheyweregiventousinthearray.WebuildtheheapbystartingattheleaflevelandtheninvokeHeapifyoneachnode.(Note:Wecannotstartatthetopofthetree.Whynot?BecausethepreconditionwhichHeapifyassumesisthattheentiretreerootedatnodeisalreadyinheaporder,exceptfor.)Actually,wecanbeabitmoreefcient.Sinceweknowthateachleafisalreadyinheaporder,wemayaswellskiptheleavesandstartwiththerstnonleafnode.Thiswillbeinposition.(Canyouseewhy?)Hereisthecode.Sincewewillworkwiththeentirearray,theparameterforHeapify,whichindicatesthecurrentheapsizewillbeequalto,thesizeofarray,inallthecalls. BuildHeap(intn,arrayA[1..n]){//buildheapfromA[1..n]fori=n/2downto1{Heapify(A,i,n) AnexampleofBuildHeapisshowninFigure7.3onpage146ofCLR.SinceeachcalltoHeapifytakestime,andwemakeroughlycallstoit,thetotalrunningtimeis2)log.Nexttimewewillshowthatthisactuallyrunsfaster,andinfactitrunsinWecannowgivetheHeapSortalgorithm.Theideaisthatweneedtorepeatedlyextractthemaximumitemfromtheheap.Aswementionedearlier,thiselementisattherootoftheheap.Butonceweremoveitweareleftwithaholeinthetree.Toxthiswewillreplaceitwiththelastleafinthetree(theoneatpositionpositionm]).Butnowtheheaporderwillverylikelybedestroyed.SowewilljustapplyHeapifytotheroottoxeverythingbackup. HeapSort(intn,arrayA[1..n]){//sortA[1..n]BuildHeap(n,A)//buildtheheapm=n//initiallyheapcontainsallwhile(m�=2){swapA[1]withA[m]//extractthem-thlargestm=m-1//unlinkA[m]fromheap LectureNotesCMSC251Heapify(A,1,m)//fixthingsup AnexampleofHeapSortisshowninFigure7.4onpage148ofCLR.WemakecallstoHeapify,eachofwhichtakestime.Sothetotalrunningtimeis1)logLecture14:HeapSortAnalysisandPartitioning(Thursday,Mar12,1998)Chapt7and8inCLR.Thealgorithmwepresentforpartitioningisdifferentfromthetexts.HeapSortAnalysis:LasttimewepresentedHeapSort.Recallthatthealgorithmoperatedbyrstbuildingaheapinabottom-upmanner,andthenrepeatedlyextractingthemaximumelementfromtheheapandmovingittotheendofthearray.Onecleveraspectofthedatastructureisthatitresidesinsidethearraytobesorted.WearguedthatthebasicheapoperationofHeapifyrunsintime,becausetheheaphaslevels,andtheelementbeingsiftedmovesdownonelevelofthetreeafteraconstantamountofwork.Basedonthiswecanseethat(1)thatittakestimetobuildaheap,becauseweneedtoapplyHeapifyroughlytimes(toeachoftheinternalnodes),and(2)thatittakestimetoextracteachofthemaximumelements,sinceweneedtoextractroughlyelementsandeachextractioninvolvesaconstantamountofworkandoneHeapify.ThereforethetotalrunningtimeofHeapSortisIsthistight?Thatis,istherunningtime?Theanswerisyes.Infact,laterwewillseethatitisnotpossibletosortfasterthantime,assumingthatyouusecomparisons,whichHeapSortdoes.However,itturnsoutthattherstpartoftheanalysisisnottight.Inparticular,theBuildHeapprocedurethatwepresentedactuallyrunsintime.AlthoughinthewidercontextoftheHeapSortalgorithmthisisnotsignicant(becausetherunningtimeisdominatedbytheextractionNonethelesstherearesituationswhereyoumightnotneedtosortalloftheelements.Forexample,itiscommontoextractsomeunknownnumberofthesmallestelementsuntilsomecriterion(dependingontheparticularapplication)ismet.Forthisreasonitisnicetobeabletobuildtheheapquicklysinceyoumaynotneedtoextractalltheelements.BuildHeapAnalysis:LetusconsidertherunningtimeofBuildHeapmorecarefully.Asusual,itwillmakeourlivessimplebymakingsomeassumptionsabout.Inthiscasethemostconvenientassumptionisisoftheform,whereistheheightofthetree.Thereasonisthataleft-completetreewiththisnumberofnodesisacompletetree,thatis,itsbottommostlevelisfull.Thisassumptionwillsaveusfromworryingaboutoorsandceilings.Withthisassumption,level0ofthetreehas1node,level1has2nodes,anduptolevel,whichhasnodes.AlltheleavesresideonlevelRecallthatwhenHeapifyiscalled,therunningtimedependsonhowfaranelementmightsiftdownbeforetheprocessterminates.Intheworstcasetheelementmightsiftdownallthewaytotheleaflevel.Letuscounttheworkdonelevelbylevel.Atthebottommostleveltherearenodes,butwedonotcallHeapifyonanyofthesesotheworkis0.Atthenexttobottommostleveltherearenodes,andeachmightsiftdown1level.Atthe3rdlevelfromthebottomtherearenodes,andeachmightsiftdown2levels.Ingeneral,atlevel