lecture 3 heap sort and quick sort - umass amherst 241 f19 lecture 3.pdf · lecture 3 heap sort and...

Lecture3HeapSortandQuickSort

ECE241– Adv.ProgrammingIFall2019MikeZink

ECE 241 – Adv. Programming I Fall 2019 © 2019 Mike Zink

Overview

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• HeapSort• QuickSort


Objective

• Learnhowperformance ofMergesortcanbefurther improvedbyusingHeapsortandQuicksort


Heap Sort

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• Use“heap”datastructure tomanageinformation

• Makesalsoanefficientpriorityqueue (morelaterinthesemester)


Binary Heap

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ValueIndex


Binary Heap

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• Represent treeassinglelist:• Root oftreeisA[1],withindexi ofnode

• PARENT(i)return ⌊i/2⌋

• Left(i)return 2i

• Right(i)return 2i +1


Max-heap

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• A[PARENT(i)] ≥ A[i]:Valueofnode i isatmostvalueofitsparent

• Largestvaluestoredatroot• Subtreerootedatnodecontainsnovalueslagerthanvalueofnodeitself

• UsedforHeapsort


Min-Heap

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• A[PARENT(i)] ≤ A[i]• Smallestvaluestoredatroot• Subtreerootedatnodecontainsnovaluessmallerthanvalueofnodeitself


Max-Heapify

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Max-Heapify

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Max-Heapify

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Max-Heapify

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Max-Heapify(A,i)1 l ⃪ LEFT(i)

2 r ⃪ RIGHT(i)

3 ifl≤ heap-size[A]andA[l]>A[i]

4 thenlargest⃪ l

5 elselargest⃪ i

6 ifr≤ heap-size[A]andA[r]>A[largest]

7 thenlargest⃪ r

8 iflargest≠i

9 thenexchangeA[i]⟷ A[largest]

10 MAX-HEAPIFY(A,largest)


Building a Heap

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Build-Max-Heap(A) 1 heap-size[A]⃪ length[A]

2 forI⃪ ⌊length[A]/2⌋ downto 1

3 doMAX-HEAPIFY(A,i)

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Building a Heap: Example

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Heapsort Algorithm

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HeapSort(A) 1 BUILD-MAX-HEAP(A)

2 fori ⃪ length[A]downto 2

3 doexchangeA[1]⟷ A[i]

4 heap-size[A]⃪ heap-size[A]– 1

5 MAX-HEAPIFY(A,1)


Heapsort Algorithm: Example (i)

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Heapsort Algorithm: Example (ii)

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Heapsort Algorithm: Example (iii)

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Heapsort Algorithm: Example (iv)

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Heapsort Algorithm: Example (v)

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Heapsort Algorithm: Example (vi)

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Heapsort Algorithm: Example (vii)

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Heapsort Algorithm: Example (viii)

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Heapsort Algorithm: Example (iv)

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Heapsort Algorithm: Example (x)

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Heapsort – Analysis

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• Heapsort takestimeO(nlogn)• BUILD-MAX-HEAP takestimeO(n)• MAX-HEAPIFY takestimeO(logn)


Quick Sort

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• UsesdivideandconquerasinMergeSort• NoADDITIONAL storageusage=>overcomesMergeSortweakness

• Trade-off: Performance diminished iflistcanbedividedinhalf


Quick Sort – Pivot Value

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1. SelectPivotValue:• Simplyfirstiteminlist

• Assistwithsplittinglist• Splitpoint usedtodividelistforsubsequent calls


Quick Sort – Partitioning

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2. Partitioning:• Setleftmark andrightmark

• Moveitemsto”right”sideofpivotvalue

• Convergesplitpoint


Quick Sort – Exchange Pivot

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3. ExchangePivot:• Stopwhenrightmark <=leftmark• Moveitemsto”right”sideofpivotvalue• Convergesplitpoint• QuickSortforleft- andrighthalf


Quick Sort – Implementation

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def quickSort(alist):quickSortHelper(alist,0,len(alist)-1)

def quickSortHelper(alist,first,last):if first<last:

splitpoint = partition(alist,first,last)

quickSortHelper(alist,first,splitpoint-1)quickSortHelper(alist,splitpoint+1,last)



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def partition(alist,first,last):pivotvalue = alist[first]

leftmark = first+1rightmark = last

done = Falsewhile not done:

while leftmark <= rightmark and alist[leftmark] <= pivotvalue: leftmark = leftmark + 1

while alist[rightmark] >= pivotvalue and rightmark >= leftmark:rightmark = rightmark -1

if rightmark < leftmark: done = True else: temp = alist[leftmark]

alist[leftmark] = alist[rightmark]alist[rightmark] = temp

temp = alist[first]alist[first] = alist[rightmark]alist[rightmark] = temp

return rightmark



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alist = [54,26,93,17,77,31,44,55,20]quickSort(alist)print(alist)


Quick Sort – Analysis

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• logndivisions, ifpartitionsoccuralwaysinmiddleoflist

• nitemstocheckagainstpivotvaluetofindsplitpoint• =>nlogn• Noadditionalmemory requirements


Quick Sort – Analysis (Worst Case)

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• Splitpointsnotinthemiddle,skewed toleftorright• =>sortingalistof0andn-1 items,then0andn-2,andsoon

• Results inO(n2)plusoverhead requiredbyrecursion• Medianofthethree:pivotvalueismedianvalueoffirst,middle,andlastelement=>chooseabetter“middlevalue”


Next Steps

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• NextlectureonThursday• DiscussiononThursday• Homework1dueonThursday at11PM

lecture 3 heap sort and quick sort - umass amherst 241 f19 lecture 3.pdf · lecture 3 heap sort and...

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