1. Heap Data Structure The heap data structure is an array object that can be viewed as a nearly complete binary tree.

2. Heap 1 2 3 4 5 6 7 8 9 10 (a) 16 2 9 8 3 10 7 14 4 1

3. Heap (b) 1 4 2 3 9 7 8 10 14 16 5 4 3 2 1 10 9 8 7 6

5. Maintaining the Heap Property The function of MAX-HEAPIFY is to let the value at A[i] “float down” in the max-heap so that the sub-tree rooted at index i becomes a max-heap

6. Maintaining the Heap Property MAX-HEAPIFY(A, i) l  LEFT(i) r  RIGHT(i) if l <= heap-size[A] and A[l] >A [i] then largest  l else largest  i

7. Maintaining the Heap Property If r<= heap-size[A] and A[r] > A[largest] then largest  r If largest ≠ i then exchange A[i]  A[largest] MAX-HEAPIFY(A, largest)

8. Maintaining the Heap Property 1 2 3 4 5 6 7 8 9 10 i (a) 16 2 9 14 3 10 7 4 8 1

9. Maintaining the Heap Property 1 2 3 4 5 6 7 8 9 10 i (b) 16 2 9 4 3 10 7 14 8 1

10. Maintaining the Heap Property 1 2 3 4 5 6 7 8 9 10 i (c) 16 2 9 6 3 10 7 14 4 1

11. Building a Heap BUILD-MAX-HEAP(A) heap-size[A]  length[A] for i  length[A]/2 downto 1 do MAX-HEAPIFY(A, i)

12. Building a Heap 3 2 16 1 10 4 7 8 14 9 A

13. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (a) 4 14 9 2 10 3 16 1 8 7

14. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (b) 4 14 9 2 10 3 16 1 8 7

15. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (c) 4 2 9 14 10 3 16 1 8 7

16. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (d) 4 2 9 14 3 10 16 1 8 7

17. Building a Heap 1 2 3 4 5 6 7 8 9 10 i (e) 4 2 9 14 3 10 7 16 8 1

18. Building a Heap 1 2 3 4 5 6 7 8 9 10 (f) 16 2 9 8 3 10 7 14 4 1

19. Heapsort Algorithm HEAPSORT(A) BUILD-MAX-HEAP(A) for I  length[A] downto 2 do exchange A[1]   A[i] heap-size[A]  heap-size[A] -1 MAX-HEAPIFY(A,1)

20. Heapsort Algorithm (a) 16 2 9 8 3 10 7 14 4 1

21. Heapsort Algorithm (b) i 14 2 9 4 3 10 7 8 1 16

22. Heapsort Algorithm (c) i 10 2 1 4 3 9 7 8 16 14

23. Heapsort Algorithm (d) i 9 1 4 2 3 7 8 16 14 10

24. Heapsort Algorithm (e) i 8 1 4 3 2 7 16 14 10 19

25. Heapsort Algorithm (f) i 7 1 3 2 4 16 14 10 9 8

26. Heapsort Algorithm (g) i 4 1 3 2 16 14 10 9 8 7

27. Heapsort Algorithm (h) i 3 1 2 16 14 10 9 8 7 4

28. Heapsort Algorithm (i) i 2 1 16 14 10 9 8 7 4 3

29. Heapsort Algorithm (j) i 1 2 16 14 10 9 8 7 4 3

30. Heapsort Algorithm k 3 4 7 2 9 1 16 14 10 8 A