Priority queues
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Priority Queues PowerPoint PPT Presentation


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Priority Queues. What is a Priority Queue?. Container of elements where each element has an associated key A key is an attribute that can identify rank or weight of an element Examples – passenger, todo list. Priority Queue ADT Operations. size() isEmpty()

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Priority Queues

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Priority Queues


What is a Priority Queue?

  • Container of elements where each element has an associated key

  • A key is an attribute that can identify rank or weight of an element

    • Examples – passenger, todo list


Priority Queue ADT Operations

  • size()

  • isEmpty()

  • insertItem(k, e) – insert element e with key k

  • minElement() – return ref to min element

  • minKey() – return const ref to smallest key

  • removeMin() – remove and return element with smallest key


insertItem(5, A)

insertItem(9, C)

insertItem(3, B)

insertItem(7, D)

minElement()

minKey()

removeMin()

size()

minElement()

removeMin()

removeMin()

removeMin()

removeMin()

isEmpty()

Example


Implementation

  • Using an array?

  • Using a linked list?

  • Using a binary search tree?

  • Running time of insertItem/removeMin?


Heaps

  • A heap is a priority queue implemented with a binary tree


Heaps

  • Heap-Order Property: In a heap T, for every node v other than the root, the key stored at v is greater than or equal to the key stored at v’s parent


Heaps

  • Complete Binary Tree Property: A heap T with height h is a complete binary tree, that is, levels 0,1,2,…,h-1of T have the maximum number of nodes possible and all the internal nodes are to the left of the external nodes in level h-1.

  • What does this give us?


Example Heap

root

4

9

7

56

10

15

11


Heap Implementation

  • Insertion algorithm

new_node

5

root

4

9

7

56

10

15

11


Up-Heap Bubbling

while new node is smaller than its parent

move parent down

  • Running time?


Heap Implementation

  • Deletion algorithm

delete 4

root

4

9

5

56

10

7

11

15


Down-Heap Bubbling

if right child is null

if left child is null

insert

else if left child is smaller than current

move left up

if both children not null

move up smallest child


Vector Implementation

  • Children are positions index*2 and index*2+1

  • Implementation of insert and remove?

0 1 2 3 4 5 6 7


Heap Sort

  • Algorithm to use a heap to sort a list of numbers

  • Running time?


BuildHeap Algorithm

  • Goal: Insert N items into initially empty heap

  • Algorithm 1: Perform N inserts

    • Worst-case running time?


Efficient BuildHeap

  • Idea: Start at level h-1 and bubble down nodes 1 at a time

    for(int i = currentSize/2; i > 0; i--)

    percolateDown(i);


Running Time

  • Running time is no more than the sum of the heights of all nodes

root

56

9

15

7

10

4

11


BuildHeap Proof

  • Theorem: For the perfect binary tree of height h containing 2h+1-1 nodes, the sum of the heights of the nodes is 2h+1-1-(h+1)

  • Proof: 1 node at height h, 2 nodes at height h-1, 22 nodes at height h-1, 2i nodes at height h-i

    h

  • Sum S = ∑ 2i(h-i)

    i=0


BuildHeap Proof

  • S = h+2(h-1)+4(h-2) +...+2h-1(1) + 2h(0)

  • 2S = 2h + 4(h-1) + 8(h-2)+ ... +2h(1)

  • 2S-S = -h + 2 + 4 + 8 + ... + 2h-1 + 2h

  • = 2h+1 -1 - 1 - h

  • = 2h+1 -1 - (h+1)

  • = < 2N (N = 2h -> 2h+1)

  • = O(N)


Event Simulation

  • Bank simulation using priority queues?


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