COMP251:Heaps&Heapsort

JérômeWaldispühl SchoolofComputerScience

McGillUniversityFrom(Cormenetal.,2002)

BasedonslidesfromD.Plaisted(UNC)

Heapdatastructure

•  Tree-baseddatastructure(here,binarytree,butwecanalsousek-arytrees)

•  Max-Heap–  Largestelementisstoredattheroot.–  forallnodesi,excludingtheroot,A[PARENT(i)]≥A[i].

•  Min-Heap–  Smallestelementisstoredattheroot.–  forallnodesi,excludingtheroot,excludingtheroot,A[PARENT(i)]≤A[i].

Heaps–Example

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Max-heapasabinarytree.

Lastrowfilledfromleftoright.

Heapsasarrays

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Max-heapasanarray.

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Mapfromarrayelementstotreenodesandviceversa•  Root–A[1]•  Lef[i]–A[2i]•  Right[i]–A[2i+1]•  Parent[i]–A[⎣i/2⎦]

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Height

•  Heightofanodeinatree:thenumberofedgesonthelongestsimplepathdownfromthenodetoaleaf.

•  Heightofaheap=heightoftheroot=Θ(lgn).•  MostBasicoperaionsonaheapruninO(lgn)ime

•  Shapeofaheap

SoringwithHeaps•  Usemax-heapsforsoring.•  Thearrayrepresentaionofmax-heapisnotsorted.•  Stepsinsoring–  Convertthegivenarrayofsizentoamax-heap(BuildMaxHeap)–  Swapthefirstandlastelementsofthearray.

•  Now,thelargestelementisinthelastposiion–whereitbelongs.•  Thatleavesn–1elementstobeplacedintheirappropriatelocaions.

•  However,thearrayoffirstn–1elementsisnolongeramax-heap.•  Floattheelementattherootdownoneofitssubtreessothatthearrayremainsamax-heap(MaxHeapify)

•  Repeatstep2unilthearrayissorted.

Heapsort•  Combinesthebekerakributesofmergesortandinserionsort.– Likemergesort,butunlikeinserionsort,runningimeisO(nlgn).

– Likeinserionsort,butunlikemergesort,sortsinplace.

•  Introducesanalgorithmdesigntechnique– Createdatastructure(heap)tomanageinformaionduringtheexecuionofanalgorithm.

•  Theheaphasotherapplicaionsbesidesoring.– PriorityQueues(SeeCOMP250)

Maintainingtheheapproperty•  Supposetwosubtreesaremax-heaps,buttherootviolatesthemax-heapproperty.

•  Fixtheoffendingnodebyexchangingthevalueatthenodewiththelargerofthevaluesatitschildren.–  Theresulingtreemayhaveasubtreethatisnotaheap.

•  Recursivelyfixthechildrenunilallofthemsaisfythemax-heapproperty.

MaxHeapify–Example26

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14MaxHeapify(A,2)

Noden=2

ProcedureMaxHeapify

MaxHeapify(A, i, n)1. l ← leftNode(i)2. r ← rightNode(i)3. if l ≤ heap-size[A] and A[l]>A[i]4. then largest ← l5. else largest ← i6. if r ≤ n and A[r]>A[largest]7. then largest ← r8. if largest ≠ i9. then exchange A[i] ↔ A[largest]10. MaxHeapify(A, largest)

Assump&on:Lef(i)andRight(i)aremax-heaps. nisthesizeoftheheap.

Timetofixnodeianditschildren=Θ(1)

Timetofixthesubtreerootedatoneofi’schildren=T(sizeofsubtree)

WorstcaserunningimeofMaxHeapify(A,n)

•  T(n)=T(largest)+Θ(1)

•  largest≤2n/3(worstcaseoccurswhenthelastrowoftreeisexactlyhalffull)

•  T(n)≤T(2n/3)+Θ(1)⇒T(n)=O(lgn)

•  Alternately,MaxHeapifytakesO(h)wherehistheheightofthenodewhereMaxHeapifyisapplied

≤2 ⋅n3

≤n3

Ο(log(n))

WorstcaserunningimeofMaxHeapify(A,n)

Buildingaheap

•  UseMaxHeapifytoconvertanarrayAintoamax-heap.•  CallMaxHeapifyoneachelementinabokom-upmanner.

BuildMaxHeap(A)1. n ← length[A]2. for i ← ⎣length[A]/2⎦ downto 13. do MaxHeapify(A, i, n)

BuildMaxHeap–Example

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InputArray:

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Staringtree(notmax-heap)

BuildMaxHeap–Example

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MaxHeapify(4)

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MaxHeapify(3)

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MaxHeapify(2)

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MaxHeapify(1)

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CorrectnessofBuildMaxHeap•  LoopInvariant:Atthestartofeachiteraionoftheforloop,

eachnodei+1,i+2,…,nistherootofamax-heap.•  Ini&aliza&on:–  Beforefirstiteraioni=⎣n/2⎦–  Nodes⎣n/2⎦+1,⎣n/2⎦+2,…,nareleaves,hencerootsoftrivialmax-heaps.

•  Maintenance:–  ByLI,subtreesatchildrenofnodeiaremaxheaps.–  Hence,MaxHeapify(i)rendersnodeiamaxheaproot(whilepreservingthemaxheaprootpropertyofhigher-numberednodes).

–  Decremeningireestablishestheloopinvariantforthenextiteraion.

RunningTimeofBuildMaxHeap•  Looseupperbound:–  CostofaMaxHeapifycall×No.ofcallstoMaxHeapify–  O(lgn)×O(n)=O(nlgn)

•  Tighterbound:–  CostofMaxHeapifyisO(h).–  ≤⎡n/2h+1⎤nodesofheighth.–  Heightofheapis

n2h+1

!

""#

$$h=0

lgn"% $&

∑ O(h) =O n h2h

h=0

lgn"% $&

∑(

)**

+

,--=O(n)

lgn!" #$

RunningimeofBuildMaxHeapisO(n)

Heapsort

1.  Buildsamax-heapfromthearray.

2.  Putthemaximumelement(i.e.theroot)atthecorrectplaceinthearraybyswappingitwiththeelementinthelastposiioninthearray.

3.  “Discard”thislastnode(knowingthatitisinitscorrectplace)bydecreasingtheheapsize,andcallMAX-HEAPIFYonthenewroot.

4.  Repeatthisprocess(goto2)unilonlyonenoderemains.

Heapsort(A)

HeapSort(A)1. Build-Max-Heap(A)2. for i ← length[A] downto 23. do exchange A[1] ↔ A[i] 4. MaxHeapify(A, 1, i-1)

Heapsort–Example

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Heapsort–Example

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Heapify

Heapsort–Example

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Heapify

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Heapsort–Example

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Heapsort–Example

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HeapProceduresforSoring

•  BuildMaxHeapO(n)•  forloopn-1imes(i.e.O(n))– exchangeelementsO(1)– MaxHeapifyO(lgn)

=>HeapSortO(nlgn)