CSE326:DataStructures Lecture#4 HeapsmorePriorityQs BartNiswonger - - PDF document

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CSE326:DataStructures Lecture#4 HeapsmorePriorityQs

BartNiswonger SummerQuarter2001

Today’sOutline

• Returnquizzes
• ThingsBartDidn’tFinishonFriday

(insert&d-Heaps)

• LeftistHeaps
• SkewHeaps
• ComparingHeaps

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PriorityQueueADT

• PriorityQueueoperations

– create – destroy – insert – deleteMin – is_empty

• PriorityQueueproperty:fortwoelementsin

thequeue,x andy,ifx hasalowerpriority value thany,x willbedeletedbeforey

F(7) E(5) D(100) A(4) B(6)

insert deleteMin

G(9) C(3) 20 14 12 9 11 8 10 6 7 5 4 2 2 4 5 7 6 10 8 11 9 12 14 20 12

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

NiftyStorageTrick

• Calculations:

– child: – parent: – root: – nextfree:

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DeleteMin

20 14 12 9 11 8 10 6 7 5 4 ? 2 20 14 12 9 11 8 10 6 7 5 4 2 pqueue.deleteMin()

Insert

20 14 12 9 11 8 10 6 7 5 4 2 20 14 12 9 11 8 10 6 7 5 4 2 pqueue.insert(3) ?

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PercolateUp

20 14 12 9 11 8 10 6 7 5 4 2 ? 20 14 12 9 11 8 ? 6 7 5 4 2 10 20 14 12 9 11 8 5 6 7 ? 4 2 10 20 14 12 9 11 8 5 6 7 3 4 2 10 3 3 3

InsertCode

voidinsert(Objecto){ assert(!isFull()); size++; newPos= percolateUp(size,o); Heap[newPos]=o; }

intpercolateUp(inthole, Objectval){ while(hole>1&& val<Heap[hole/2]) Heap[hole]=Heap[hole/2]; hole/=2; } returnhole; }

runtime:

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OtherPriorityQueue Operations

• decreaseKey

– giventhepositionofanobjectinthequeue, reduceitspriorityvalue

• increaseKey

– giventhepositionofananobjectinthequeue, increaseitspriorityvalue

• remove

– giventhepositionofanobjectinthequeue, removeit

• buildHeap

– givenasetofitems,buildaheap

DecreaseKey,IncreaseKey, andRemove

voiddecreaseKey(int obj){ assert(size>=obj); temp=Heap[obj]; newPos= percolateUp(obj,temp); Heap[newPos]=temp; } voidincreaseKey(intobj){ assert(size>=obj); temp=Heap[obj]; newPos=percolateDown(obj,temp); Heap[newPos]=temp; }

voidremove(intobj){ assert(size>= obj); percolateUp(obj, NEG_INF_VAL); deleteMin(); }

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BuildHeap

Floyd’sMethod.Thankyou,Floyd.

5 11 3 10 6 9 4 8 1 7 2 12 pretendit’saheapandfixtheheap-orderproperty! 2 7 1 8 4 9 6 10 3 11 5 12

Build(this)Heap

6 7 1 8 4 9 2 10 3 11 5 12 6 7 10 8 4 9 2 1 3 11 5 12 11 7 10 8 4 9 6 1 3 2 5 12 11 7 10 8 4 9 6 5 3 2 1 12

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Finally…

11 7 10 8 12 9 6 5 4 2 3 1 runtime:

ThinkingaboutHeaps

• Observations

– findingachild/parentindexisamultiply/divideby two – operationsjumpwidelythroughtheheap – eachoperationlooksatonlytwonewnodes – insertsareatleastascommonasdeleteMins

• Realities

– divisionandmultiplicationbypowersoftwoare fast – lookingatonenewpieceofdatasucksinacache line – withhuge datasets,diskaccessesdominate

8 4 9 6 5 4 2 3 1 8 10 12 7 11

Solution:d-Heaps

• Eachnodehasd children
• Stillrepresentableby

array

• Goodchoicesford:

– optimizeperformancebasedon #ofinserts/removes – chooseapoweroftwofor efficiency – fitonesetofchildreninacache line – fitonesetofchildrenona memorypage/diskblock

3 7 2 8 5 121110 6 9 1 12

OneMoreOperation

• Mergetwoheaps.Ideas?

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Merge

Giventwoheaps,mergethemintoone heap

– firstattempt:inserteachelementofthe smallerheapintothelarger. runtime: – secondattempt:concatenateheaps’arrays andrunbuildHeap. runtime:

HowaboutO(logn)time?

Idea:HangaNewTree

12 13 10 6 11 5 2

+

10 14 4 9 1

=

14 12 13 10 6 4 9 11 5 1 2 ? 10 Now,just percolatedown!

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Idea:HangaNewTree

12 13 10 6 11 5 2

+

10 6 11 5 2

=

12 13

Problem?

12 13 12 13

LeftistHeaps

• Idea:
• Leftistheap:

– almostallnodesareontheleft – allthemergingworkisontheright

makeitsothatalltheworkyou havetodoinmaintainingaheap isinonesmallpart

11 thenullpathlength(npl) ofanodeisthenumber

• fnodesbetweenitandanullinthetree

RandomDefinition: NullPathLength

• npl(null)=-1
• npl(leaf)=0
• npl(single-child

node)=0

1 1 1 2 anotherwayoflookingatit: nplistheheightofcomplete subtreerootedatthisnode

LeftistHeapProperties

• Heap-orderproperty

– parent’spriorityvalueis tochildrens’ priority values – result:minimumelementisattheroot

• Leftistproperty

– nullpathlengthofleftsubtree is nplofright subtree – result:treeisatleastas“heavy”ontheleftas theright

Areleftisttreescomplete?Balanced?

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Leftisttreeexamples

NOT leftist leftist 1 1 1 2 1 1 2 1 1 leftist everysubtreeofaleftist treeisleftist,comrade!

RightPathinaLeftistTreeisShort

• Iftherightpathhaslengthatleast

r,thetreehasatleast2r-1 nodes

• Proofbyinduction

Basis:r=1.Treehasatleast

• nenode:21 - 1=1

Inductivestep:assumetrueforr’<r.Therightsubtreehasa rightpathofatleastr- 1nodes,soithasatleast2r- 1 - 1 nodes.Theleftsubtreemustalsohavearightpathofatleast r- 1 (otherwise,thereisanullpathofr- 3,lessthanthe rightsubtree).Again,thelefthas2r- 1 - 1 nodes.Alltoldthen, thereareatleast: 2r- 1 - 1+2r- 1 - 1+1=2r - 1

• So,aleftisttreewithatleastn nodeshasaright

pathofatmostlogn nodes

1 1 2 1

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Whew!

ToDo

• UnixdevelopmentTutorial

– Tuesday– 10:50am– Sieg322

• FinishProjectIforWednesday
• Readchapters1&2

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ComingUp

• Theory!
• ProofbyInduction
• AsymptoticAnalysis
• Quiz#2(Thursday)