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

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

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

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

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

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

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

8. upHeap Code
void PQ::upHeap (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; }

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

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

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

12. DownHeap (helper) void PQ::downHeap (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

13. Heaps  O(N*lg (N)) sort in worst case
Heap Sort Overview
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
1) Heapify vector v
STL assistance
Roll our own buildHeap
2) While (N != 1)
Swap largest element and back of v
N--
Fix up heap property at root (downHeap)

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

15. buildHeap Start at lowest internal node: position = ? Then do
downHeap (position)

16. buildHeap Cont’d

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

18. 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--)
downHeap (i);
while (N != 1) // 2 or more elements {
swap (v[0], v[N-1]);
N--;
downHeap (0); }