Lecture 13 Heaps Container Adapters priorityqueue impl with

Lecture 13 – Heaps Container Adapters priority_queue – impl. with heap Max and Min Heaps push () pop () Heap Sort Heapifying (make_heap) 1

l Priority Queues Recall: 3 Container Adapters – – – l Stack and Queue – – l Underlying default sequence container: deque Deque: efficient operations on front and back Priority Queue – – – 2 Stack Queue Priority Queue – implementation? Uses <vector> – efficient only at back Maintains complete tree w/special ordering property Vector w/property is Heap

Complete Binary Tree All levels full except last Last level children to left Why store tree in vector? parent (i) = p (i) = ? left. Child (i) = lc (i) = ? right. Child (i) = rc (i) = ? 3

Max and Min Heaps Heap property (Max): Value (X) >= Value (Child (X)) 4

priority_queue: : push 1) v. push_back (50); 5

Upheap (restore heap property) 2) up. Heap (v. size () – 1); 6

Push Code void PQ: : push (const T& item) { v. push_back (item); up. Heap (v. size ( ) – 1); } 7

up. Heap Code void PQ: : up. Heap (size_t pos) { T item = v[pos]; size_t i; // Move parent down, item up for (i = pos; i != 0 && item > v[p (i)]; i = p (i)) { v[i] = v[p (i)]; // note: swap is unnecessary v[i] = item; } 8

priority_queue: : pop 18 v[0] = v. back (); v. pop_back (); 9

Downheap Compare v[i] with max child Move max child up if necessary 10

Pop void PQ: : pop () { // Overwrite top element with last v[0] = v. back (); v. pop_back (); // O (1) // Move elem down to proper place down. Heap (0); } 11

Down. Heap (helper) 12 void PQ: : down. Heap (size_t pos) { // Move “v[pos]” down, max child up size_t i, mc; // mc is index of max child T val = v[pos]; for (i = pos; (mc = lc (i)) < v. size (); i = mc) { if (mc + 1 < v. size () && v[mc] < v[mc + 1]) ++mc; // “mc” is now referencing larger child // Compare val and ? // Move child up if necessary } // Place val in correct spot }

Heap Sort Overview l Heaps O(N*lg (N)) sort in worst case – – – Push all elements Retrieve and place in array back to front Naïve: 2*Sum (i =1: N) [lg(i)] OR l 1) Heapify vector v – – l 2) While (N != 1) – – – 13 STL assistance Roll our own build. Heap Swap largest element and back of v N-Fix up heap property at root (down. Heap)

Make_Heap (STL) int arr[] = { 50, 20, 75, 35, 25 }; make_heap (arr, arr + 5); // in STL // Details later on impl. - <algorithm> 14

build. Heap Start at lowest internal node: position = ? Then do down. Heap (position) 15

build. Heap Cont’d 16

Implementing Heap Sort (Swap and Fix up) Swap front and back Do down. Heap (0) 17

Heap Sort (Details) // Create heap from vector bottom-up // Start with lowest internal node // N is taken as size of ‘v’ // ‘for’ creates heap bottom-up: O(N) for (i = (N – 2) / 2; i >= 0; i--) down. Heap (i); while (N != 1) // 2 or more elements { swap (v[0], v[N-1]); N--; down. Heap (0); } 18