Priority Queues

• 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

What is a Priority Queue?

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

Example

• insertItem(5, A)• insertItem(9, C)• insertItem(3, B)• insertItem(7, D)• minElement()• minKey()• removeMin()

• size()• minElement()• removeMin()• removeMin()• removeMin()• removeMin()• isEmpty()

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

7 9

4

56 101115

root

Heap Implementation

• Insertion algorithm

7 9

4

56 101115

root

5new_node

Up-Heap Bubbling

while new node is smaller than its parentmove parent down

• Running time?

Heap Implementation

• Deletion algorithm

5 9

4

56 10117

root

15

delete 4

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

15 9

56

7 10114

root

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?