Lecture 18 Heaps and Priority Queues - visionlab.engr.ccny...

CSC212 Data Structure

- Section FG

Lecture18HeapsandPriorityQueues

Instructor:FengHUDepartmentofComputerScience

CityCollegeofNewYork

Heaps

•Chapter11hasseveralprogrammingprojects,includingaprojectthatusesheaps.

• Thispresentationshowsyouwhataheapis,anddemonstratestwooftheimportantheapalgorithms.

Data Structuresand Other ObjectsUsing C++

Topics

•HeapDefinition•HeapApplications

• priorityqueues(chapter8),sorting(chapter13)• TwoHeapOperations– add,remove

• reheapificationupwardanddownward• whyisaheapgoodforimplementingapriorityqueue?

•HeapImplementation• usingbinary_tree_nodeclass• usingfixedsizeordynamicarrays

HeapsDefinition

Aheapisacertainkindofcompletebinarytree.

Heaps

Aheap isacertainkindofcompletebinarytree.

When a completebinary tree is built,

its first node must bethe root.

Root

Heaps

Completebinarytree. Left childof theroot

The second node isalways the left child

of the root.

Heaps

Completebinarytree. Right childof the

root

The third node isalways the right child

of the root.

Heaps

Completebinarytree.

The next nodesalways fill the next

level from left-to-right.

Heaps

Completebinarytree.

Heaps


Each node in a heapcontains a key thatcan be compared toother nodes' keys.

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Heaps


The "heap property"requires that each

node's key is >= thekeys of its children

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Whatitisnot:ItisnotaBST

• Inabinarysearchtree,theentriesofthenodescanbecomparedwithastrictweakordering.Tworulesarefollowedforeverynoden:

• TheentryinnodenisNEVERlessthan anentryinitsleftsubtree• Theentryinthenodenislessthan everyentryinitsright subtree.

• BSTisnotnecessarilyacompletetree

Whatitis:HeapDefinition

• Aheapisabinarytreewheretheentriesofthenodescanbecomparedwiththelessthan operatorofastrictweakordering.Inaddition,tworulesarefollowed:

• TheentrycontainedbythenodeisNEVERlessthan theentriesofthenode’schildren

• ThetreeisaCOMPLETEtree.

• Q:whereisthelargestentry?à forwhat....

Application:PriorityQueues

• Apriorityqueueisacontainerclassthatallowsentriestoberetrievedaccordingtosomespecificprioritylevels

• Thehighestpriorityentryisremoved first• Ifthereareseveralentrieswithequallyhighpriorities, thenthepriorityqueue’s implementationdetermineswhichwillcomeoutfirst(e.g.FIFO)

• Heapissuitableforapriorityqueue

ThePriorityQueueADTwithHeaps

• Theentrywiththehighestpriorityisalwaysattherootnode

• Focusontwopriorityqueueoperations• addinganewentry• removetheentrywiththehighestpriority

• Inbothcases,wemustensurethetreestructureremainstobeaheap• wearegoingtoworkonaconceptualheapwithoutworryingaboutthepreciseimplementation

• laterIamgoingtoshowyouhowtoimplement...

AddingaNodetoaHeap

❶ Putthenewnodeinthenextavailablespot.

❷ Pushthenewnodeupward,swappingwithitsparentuntilthenewnodereachesanacceptablelocation.

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AddingaNodetoaHeap



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AddingaNodetoaHeap



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AddingaNodetoaHeap

✔Theparenthasakeythatis>=newnode,or

✔Thenodereachestheroot.➚ Theprocessofpushingthe

newnodeupwardiscalledreheapificationupward. 19

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Note: Note: we need to easily go from child to parent as well as parent to child.

RemovingtheTopofaHeap

❶Movethelastnodeontotheroot.

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❷ Pushtheout-of-placenodedownward,swappingwithitslargerchilduntilthenewnodereachesanacceptablelocation. 19

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✔Thechildrenallhavekeys<=theout-of-placenode,or

✔Thenodereachestheleaf.

➘ Theprocessofpushingthenewnodedownwardiscalledreheapificationdownward.

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PriorityQueuesRevisited

• Apriorityqueueisacontainerclassthatallowsentriestoberetrievedaccordingtosomespecificprioritylevels

• Thehighestpriorityentryisremoved first• Ifthereareseveralentrieswithequallyhighpriorities,thenthepriorityqueue’simplementationdetermineswhichwill comeoutfirst(e.g.FIFO)

• Heapissuitableforapriorityqueue

AddingaNode:samepriority



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45*




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45*

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✔Theparenthasakeythatis>=newnode,or

✔Thenodereachestheroot.➚ Theprocessofpushingthe

newnodeupwardiscalledreheapificationupward. 19

4222135

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45

45*

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Note: Implementation determines which 45 will be in the root, and will come out first when popping.


✔Thechildrenallhavekeys<=theout-of-placenode,or

✔Thenodereachestheleaf.➘ Theprocessofpushingthe

newnodedownwardiscalledreheapificationdownward. 19

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45*

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Note: Implementation determines which 45 will be in the root, and will come out first when popping.

HeapImplementation

• Usebinary_tree_nodeclass• node implementation isforageneralbinarytree• butwemayneedtohavedoubly linkednode

• Usearrays(page475)• Aheapisacompletebinarytree• whichcanbeimplementedmoreeasilywithanarraythanwiththenodeclass

• anddotwo-waylinks

Formulasforlocationchildrenandparentsinanarrayrepresentation

• Rootatlocation[0]• Parentofthenodein[i]isat[(i-1)/2]• Childrenofthenodein[i](ifexist)isat[2i+1]and[2i+2]• Test:

• completetreeof10,000nodes• parentof4999isat(4999-1)/2=2499• childrenof4999isat9999(V)and10,000(X)

ImplementingaHeap

• Wewillstorethedatafromthenodesinapartially-filledarray.

An array of data

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ImplementingaHeap

• Datafromtherootgoesinthe firstlocationofthearray.

An array of data

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ImplementingaHeap

• Datafromthenextrowgoes inthenexttwoarraylocations.

An array of data

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ImplementingaHeap


An array of data

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ImplementingaHeap


An array of data

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We don't care what's inthis part of the array.

ImportantPointsabouttheImplementation

• Thelinksbetweenthetree'snodesarenot actuallystoredaspointers,orinanyotherway.

• Theonlywaywe"know"that"thearrayisatree"isfromthewaywemanipulatethedata.

An array of data

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ImportantPointsabouttheImplementation

• Ifyouknowtheindexofanode,thenitiseasytofigureouttheindexesofthatnode'sparentandchildren.Formulasaregiveninthebook.

[0] [1] [2] [3] [4]

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Formulasforlocationchildrenandparentsinanarrayrepresentation

• Rootatlocation[0]• Parentofthenodein[i]isat[(i-1)/2]• Childrenofthenodein[i](ifexist)isat[2i+1]and[2i+2]• Test:

• completetreeof10,000nodes• parentof4999isat(4999-1)/2=2499• childrenof4999isat9999(V)and10,000(X)

WrapUp...

• Canyouimplement theaddandremovewiththeknowledgeoftheseformulas?

• Add• putthenewentryinthelastlocation• Pushthenewnodeupward,swappingwithitsparentuntilthenewnodereachesanacceptablelocation

• Remove• movethelastnodetotheroot• Pushtheout-of-placenodedownward,swappingwithits largerchilduntilthenewnodereachesanacceptable location

class heap{public:

....void push(const Item& entry); // addItem& pop(); // remove the highest

private:Item data[CAPACITY];size_type used;

}

WrapUp...

• Canyouimplement theaddandremovewiththeknowledgeoftheseformulas?

• Add(in-classquiz)• putthenewentryinthelastlocation

• Pushthenewnodeupward,swappingwithitsparentuntilthenewnodereachesanacceptable location

• Remove(in-classquiz)• movethelastnodetotheroot• Pushtheout-of-placenodedownward,swappingwithitslargerchilduntilthenewnodereachesanacceptablelocation

template <class Item>class heap{public:

heap () { used = 0;}void push(const Item& entry); // addItem& pop(); // remove the highestsize_t parent (size_t k) const { return (k-1)/2;}size_t l_child (size_t k) const { return 2*k+1;}size_t r_child (size_t k) const { return 2*k+2;}

private:Item data[CAPACITY];size_type used;

}

Summary

•Aheapisacompletebinarytree,wheretheentryateachnodeisgreaterthanorequaltotheentriesinitschildren.

• Toaddanentrytoaheap,placethenewentryatthenextavailablespot,andperformareheapificationupward.

• Toremovethebiggestentry,movethelastnodeontotheroot,andperformareheapificationdownward.

THE END

Presentation copyright 1997 Addison Wesley Longman,For use with Data Structures and Other Objects Using C++by Michael Main and Walter Savitch.

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Lecture 18 Heaps and Priority Queues - visionlab.engr.ccny...

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