1. Design and Analysis of Algorithms
Dr. Maheswari Karthikeyan 09/03/2013, Lecture 6

2. HeapSort

3. HEAPSORT
Heap
- The heap data structure is an array object which can be viewed as a nearly complete binary tree.
- the two attributes of heap are length[A] and heap-size[A]
Heap as an ARRAY
- given an index i of a node,
Parent(i)
return (i/2)
Left(i)
return 2i
Right(i)
return 2i + 1 17
PARENT
LEFT CHILD
RIGHT CHILD 5 13

4. HEAPSORT
Max-heap - max-heap property is A[parent(i)] >= A[i] Min-heap - min-heap property is A[parent(i)] <= A[i]
max-heap
min-heap 17 6 5 13 12 14

5. HEAPSORT Basic Procedures used in Heapsort algorithm
MAX-HEAPIFY – maintains the heap property.
BUILD-MAX-HEAP – produces a max-heap from an unordered input array.
HEAPSORT – sorts an array in place

6. HEAPSORT
MAX-HEAPIFY (A,i) l  left(i) r  right(i) if l <= heap-size[A] and A[l] > A[i] then largest  l else largset  i 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)

7. MAX-HEAPIFY MAX-HEAPIFY( A,2) 1 4 3 2 33 1 5 6 7 4 15 16 9 19 10 9 814 7 11
MAX-HEAPIFY( A,2)

8. MAX-HEAPIFY MAX-HEAPIFY( A,4) 1 4 3 2 3 19 5 6 7 4 15 16 9 1 10 9 8 1411
MAX-HEAPIFY( A,4)

9. MAX-HEAPIFY MAX-HEAPIFY( A,8) 1 4 3 2 3 19 5 6 7 4 15 16 9 14 10 9 8 111
MAX-HEAPIFY( A,8)

10. MAX-HEAPIFY MAX-HEAPIFY( A,1 ) 1 4 3 2 9 30 5 6 7 4 24 2 8 25 10 9 820 12 22
MAX-HEAPIFY( A,1 )

11. MAX-HEAPIFY MAX-HEAPIFY( A,1 ) 1 30 3 2 9 25 5 6 7 4 24 2 8 22 10 9 820 12 4
MAX-HEAPIFY( A,1 )

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

13. Building a HEAP 4 1 3 19 13 16 9 14 7 11 1 4 3 2 3 1 5 6 7 4 13 16 9 19 10 9 8 14 7 11

14. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,5) 1 4 3 2 3 113 16 9 19 10 9 8 14 7 11
MAX-HEAPIFY( A,5)

15. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,4) 1 4 3 2 3 113 16 9 19 10 9 8 14 7 11
MAX-HEAPIFY( A,4)

16. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,3) 1 4 3 2 3 113 16 9 19 10 9 8 14 7 11
MAX-HEAPIFY( A,3)

17. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,2) 1 4 3 2 1613 3 9 19 10 9 8 14 7 11
MAX-HEAPIFY( A,2)

18. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,4) 1 4 3 2 1613 3 9 1 10 9 8 14 7 11
MAX-HEAPIFY( A,4)

19. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,1) 1 4 3 2 1613 3 9 14 10 9 8 1 7 11
MAX-HEAPIFY( A,1)

20. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,2) 1 19 3 2 1613 3 9 14 10 9 8 1 7 11
MAX-HEAPIFY( A,2)

21. Building a HEAP 4 1 3 19 15 16 9 14 7 11 MAX-HEAPIFY( A,4) 1 19 3 2 1613 3 9 4 10 9 8 1 7 11
MAX-HEAPIFY( A,4)

22. Building a HEAP 4 1 3 19 15 16 9 14 7 11 1 19 3 2 16 14 5 6 7 4 13 3 9 7 10 9 8 1 4 11
DONE

23. Building a HEAP 5 3 17 10 84 19 6 22 9
BUILD a heap on the above array

24. Building a HEAP 5 3 17 10 84 19 6 22 9 1 84 3 2 19 22 5 6 7 4 3 17 6 10 9 8 5 9

25. 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)

26. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 19 3 2 16 14 5 6 7 4 13 3 9 7 10 9 8 1 4 11 A

27. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 11 3 2 16 14 5 6 7 4 13 3 9 7 9 8 1 4 A 19

28. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 16 3 2 11 14 5 6 7 4 13 3 9 7 9 8 1 4 A 19

29. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 4 3 2 11 14 5 6 7 4 13 3 9 7 8 1 A 16 19

30. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 14 3 2 11 13 5 6 7 4 4 3 9 7 8 1 A 16 19

31. 19 14 16 7 13 3 9 1 4 11 1 1 3 2 11 13 5 6 7 4 4 3 9 7 A 14 16 19

32. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 13 3 2 11 7 5 6 7 4 4 3 9 1 A 14 16 19

33. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 13 3 2 11 7 5 6 7 4 4 3 9 1 A 14 16 19

34. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 9 3 2 11 7 5 6 4 4 3 1 A 13 14 16 19

35. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 11 3 2 9 7 5 6 4 4 3 1 A 13 14 16 19

36. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 3 3 2 9 7 5 4 4 1 A 11 13 14 16 19

37. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 9 3 2 3 7 5 4 4 1 A 11 13 14 16 19

38. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 4 3 2 3 7 4 1 A 09 11 13 14 16 19

39. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 7 3 2 3 4 4 1 A 09 11 13 14 16 19

40. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 1 3 2 3 4 A 07 09 11 13 14 16 19

41. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 4 3 2 3 1 A 07 09 11 13 14 16 19

42. HeapSort algorithm 19 14 16 7 13 3 9 1 4 11 1 3 2 1 A 04 07 09 11 13 14 16 19

43. 19 14 16 7 13 3 9 1 4 11 A 01 03 04 07 09 11 13 14 16 19

44. RUNNING TIME OF HEAPSORT
RUNNING TIME OF MAX-HEAPIFY
Running time of MAX-HEAPIFY depends on the height h of a heap i.e O(h)
Height of a heap = lg n
T(n) = O(lg n)

45. RUNNING TIME OF HEAPSORT
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)
Running time of BUILD-MAX-HEAP is O(n)
MAX-HEAPIFY is executed n-1 times with
running time O(log n)
Total Running time of Heap Sort is
O(n) + n-1(O(log n) = O(n log n)

46. RUNNING TIME OF HEAPSORT
Running time of BULID-MAX-HEAP
lg n lg n
Total cost of BULID-MAX-HEAP is ∑ n/2 h+1 O(h) = O (n ∑ h/2 h)
h= h=0 ∞
∑ h/2 h = ½ = 2. h=
(1- ½)2
Running time of BUILD-MAX-HEAP can be bounded as
lg n ∞
O (n ∑ h/2 h) = O(n ∑ h/2 h) = O (n) .
h= h=0