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Priority Queues

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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  1. Priority Queues

  2. 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

  3. 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

  4. insertItem(5, A) insertItem(9, C) insertItem(3, B) insertItem(7, D) minElement() minKey() removeMin() size() minElement() removeMin() removeMin() removeMin() removeMin() isEmpty() Example

  5. Implementation • Using an array? • Using a linked list? • Using a binary search tree? • Running time of insertItem/removeMin?

  6. Heaps • A heap is a priority queue implemented with a binary tree

  7. 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

  8. 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?

  9. Example Heap root 4 9 7 56 10 15 11

  10. Heap Implementation • Insertion algorithm new_node 5 root 4 9 7 56 10 15 11

  11. Up-Heap Bubbling while new node is smaller than its parent move parent down • Running time?

  12. Heap Implementation • Deletion algorithm delete 4 root 4 9 5 56 10 7 11 15

  13. 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

  14. Vector Implementation • Children are positions index*2 and index*2+1 • Implementation of insert and remove? 0 1 2 3 4 5 6 7

  15. Heap Sort • Algorithm to use a heap to sort a list of numbers • Running time?

  16. BuildHeap Algorithm • Goal: Insert N items into initially empty heap • Algorithm 1: Perform N inserts • Worst-case running time?

  17. 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);

  18. Running Time • Running time is no more than the sum of the heights of all nodes root 56 9 15 7 10 4 11

  19. 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

  20. 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)

  21. Event Simulation • Bank simulation using priority queues?

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