1. Design and Analysis of Algorithms Heapsort
Haidong Xue
Fall 2013, at GSU

2. Max-Heap A complete binary tree, and … Yes Yes No
every level is completely filled, except possibly the last, which is filled from left to right
Yes
Yes No

3. Max-Heap Satisfy max-heap property: parent >= children16 14 10 8 7 9 3 2
Since it is a complete tree, it can be put into an array without lose its structure information.

4. Max-Heap 16 14 10 8 7 9 3 2 1 2 3 4 5 6 7 8

5. Max-Heap Use an array as a heap For element at i:1 16
For element at i:
Parent index =parent(i)= floor(i/2);
Left child index = left(i)=2*i;
Right child index =right(i)=2*i +1
Last non-leaf node = floor(length/2) 2 3 14 10 4 5 6 7 8 7 9 3 8
i=3 2
floor(i/2)=floor(1.5)=1
2*i = 6
2*i+1=7
floor(length/2)=4 16 14 10 8 7 9 3 2 1 2 3 4 5 6 7 8

6. Max-Heapify
Input: A compete binary tree A, rooted at i, ended at t, whose left and right sub trees are max-heaps; last node index
Output: A max-heap rooted at i.
Algorithm:
MAX-HEAPIFY (A, i, t)
1. if(right(i)>t and left(i)>t) return;
2. Choose the largest node among node i, left(i), right(i) .
3. if(the largest node is not i){
m = the index of the larger node
Exchange i with the largest node
MAX-HEAPIFY (A, m, t) }

7. Max-Heapify Example 2 16 10 14 7 9 3 8

8. Heapsort for a heap Input: a max-heap in array A
Output: a sorted array A
HEAP-SORT Algorithm:
1. BUILD-MAX-HEAP(A)
2. Last node index i = A’s last node index
3. From the last element to the second in A{
exchange (i, root);
i--;
MAX-HEAPIFY(A, root, i); }

9. Heapsort example 16 14 10 8 7 9 3 2

10. Heapsort example 14 8 10 2 7 9 3 16

11. Heapsort example 10 8 9 2 7 3 14 16

12. Heapsort example 9 8 3 2 7 10 14 16

13. Heapsort example 8 7 3 2 9 10 14 16

14. Heapsort example 7 2 3 8 9 10 14 16

15. Heapsort example 3 2 7 8 9 10 14 16

16. Array -> Max-Heap Input: a array A Output: a Max-Heap A
BUILD-MAX-HEAP(A):
Considering A as a complete binary tree, from the last non-leaf node to the first one i{
MAX-HEAPIFY(A, i, A.lastIndex); }

17. Build Heap Example 14 8 16 10 7 14 8 16 10 7

18. Heapsort Input: array A Output: sorted array A Algorithm:
1. BUILD-MAX-HEAP(A)
2. Last node index i = A’s last node index
3. From the last element to the second in A{
exchange (i, root);
i--;
MAX-HEAPIFY(A, root, i); }

19. Analysis of Heapsort Input: array A Output: sorted array A Algorithm:
1. BUILD-MAX-HEAP(A)
2. Last node index i = A’s last node index
3. From the last element to the second in A{
exchange (i, root);
i--;
MAX-HEAPIFY(A, root, i); }
O(nlgn) (or O(n) see page for why)
O(n)
O(nlgn)