Priority Queues, Heaps

CS3240, L. grewe

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2

Goals

• Describe a priority queue at the logical level and implement a priority queue as a list

• Describe the shape and order property of a heap and implement a heap in a nonlinked tree representation in an array

• Implement a priority queue as a heap

• Compare the implementations of a priority queue using a heap, a linked list, and a binary search tree

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Priority Queue

Priority QueueAn ADT in which only the item with the highest priority can be accessed

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Priority Queue

Items on a priority queue are made up of

<data, priority> pairs

Can you think of how a priority queue might be implemented as

An unsorted list?

An array-based sorted list?

A linked sorted list?

A binary search tree?

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Heaps

HeapA complete binary tree, each of whose elements contains a value that is greater than or equal to the value of each of its children

Remember?Complete tree

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Heaps

Heap with letters 'A' .. 'J'

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Heaps

Another heap with letters 'A' .. 'J'

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Heaps

C

A T

treePtr

50

20

18

30

10

Are these heaps?

treePtr

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Heaps

70

60

40 30

12

8 10

treePtr

Is this a heap?

10

Heaps

70

60

40 30

12

8

treePtr

Can we use this fact to implement an efficient PQ?

Whereis the

largestelement?

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Heaps

We have immediate access to highest priority item BUT if we remove it, the structure isn't a heap

Can we "fix" the problem efficiently?

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Heaps

Shape propertyis restored. Canorder property be restored?

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Heaps

How?

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Heaps

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Heaps

template<class ItemType>

struct HeapType

{

void reheapDown(int root, int bottom);

…

ItemType* elements; // Dynamic array

int numElements;

}

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Heaps

70

0

60

1

40

3

30

4

12

2

8

5

root

Numbernodes

leftto

rightby level

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Heaps

70

0

60

1

40

3

30

4

12

2

8

5

tree[ 0 ]

[ 1 ]

[ 2 ]

[ 3 ]

[ 4 ]

[ 5 ]

[ 6 ]

70

60

12

40

30

8

elements

Use number as index

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Heaps

ReapDown(heap, root, bottom)

if heap.elements[root] is not a leaf

Set maxChild to index of child with larger value

if heap.elements[root] < heap.elements[maxChild]

Swap(heap.elements[root], heap.elements[maxChild])

ReheapDown(heap, maxChild, bottom)

What variables do we need?

Where are a node's children?

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Heaps

// IMPLEMENTATION OF RECURSIVE HEAP MEMBER FUNCTIONSvoid HeapType<ItemType>::ReheapDown(int root, int bottom)// Pre: ItemType overloads the relational operators// root is the index of the node that may violate the // heap order property// Post: Heap order property is restored between root and // bottom

{ int maxChild ; int rightChild ; int leftChild ;

leftChild = root * 2 + 1 ; rightChild = root * 2 + 2 ;

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Heaps if (leftChild <= bottom) // ReheapDown continued { if (leftChild == bottom) maxChild = leftChld; else { if (elements[leftChild] <= elements[rightChild]) maxChild = rightChild; else maxChild = leftChild; } if (elements[root] < elements[maxChild]) { Swap(elements[root], elements[maxChild]); ReheapDown(maxChild, bottom); } }}

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Heaps

What other operations might our HeapType need?

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Heaps

Add K: Where must the new node be put?

Now what?

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Heaps

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void HeapType::ReheapUp(int root, int bottom)// Pre: ItemType overloads the relational operators// bottom is the index of the node that may violate// the heap order property. The order property is // satisfied from root to next-to-last node.// Post:Heap order property is restored between root and// bottom{ int parent; if (bottom > root) { parent = ( bottom - 1 ) / 2; if (elements[parent] < elements[bottom]) { Swap(elements[parent], elements[bottom]); ReheapUp(root, parent); } }}

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Heaps

How can heaps be used to implement Priority Queues?

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Priority Queuesclass FullPQ(){};class EmptyPQ(){};template<class ItemType>class PQType{public: PQType(int); ~PQType(); void MakeEmpty(); bool IsEmpty() const; bool IsFull() const; void Enqueue(ItemType newItem); void Dequeue(ItemType& item);private: int length; HeapType<ItemType> items; int maxItems;};

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Priority Queuestemplate<class ItemType>PQType<ItemType>::PQType(int max){ maxItems = max; items.elements = new ItemType[max]; length = 0;}template<class ItemType>void PQType<ItemType>::MakeEmpty(){ length = 0;}template<class ItemType>PQType<ItemType>::~PQType(){ delete [] items.elements;}

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Priority Queues

DequeueSet item to root element from queueMove last leaf element into root positionDecrement lengthitems.ReheapDown(0, length-1)

EnqueueIncrement lengthPut newItem in next available positionitems.ReheapUp(0, length-1)

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Priority Queues

template<class ItemType>void PQType<ItemType>::Dequeue(ItemType& item){ if (length == 0) throw EmptyPQ(); else { item = items.elements[0]; items.elements[0] = items.elements[length-1]; length--; items.ReheapDown(0, length-1); }}

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Priority Queues

template<class ItemType>void PQType<ItemType>::Enqueue(ItemType newItem){ if (length == maxItems) throw FullPQ(); else { length++; items.elements[length-1] = newItem; items.ReheapUp(0, length-1); }}

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Comparison of Priority Queue Implementations

 

  Enqueue Dequeue

Heap O(log2N) O(log2N)

Linked List O(N) O(N)

Binary Search Tree

   

Balanced O(log2N) O(log2N)

Skewed O(N) O(N)