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()

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Priority Queues

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Priority queues

Priority Queues


What is a priority queue

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

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()

Example


Implementation

Implementation

  • Using an array?

  • Using a linked list?

  • Using a binary search tree?

  • Running time of insertItem/removeMin?


Heaps

Heaps

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


Heaps1

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


Heaps2

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

Example Heap

root

4

9

7

56

10

15

11


Heap implementation

Heap Implementation

  • Insertion algorithm

new_node

5

root

4

9

7

56

10

15

11


Up heap bubbling

Up-Heap Bubbling

while new node is smaller than its parent

move parent down

  • Running time?


Heap implementation1

Heap Implementation

  • Deletion algorithm

delete 4

root

4

9

5

56

10

7

11

15


Down heap bubbling

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

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

Heap Sort

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

  • Running time?


Buildheap algorithm

BuildHeap Algorithm

  • Goal: Insert N items into initially empty heap

  • Algorithm 1: Perform N inserts

    • Worst-case running time?


Efficient buildheap

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

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

root

56

9

15

7

10

4

11


Buildheap proof

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 proof1

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

Event Simulation

  • Bank simulation using priority queues?


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