Sorting with some help from heaps.

1. CS 321. Algorithm Analysis & Design
Lecture 4
Heap Sort

2. 4 9 2 1 8 5 6

3. 4 9 2 1 8 5 6

4. 4 9 2
1
8 5 6

5. 4 9 2
1
8 5 6

6. 4 9
21
8 5 6

7. 4 9
21
8 5 6

8. 4
9
21
8 5 6

9. 4
9
21
8 5 6

10. 4
9
21
8
5
6

11. 4
9
21
8
5
6

12. 4
9
21
8
5 6

13. 4
9
21
8
5 6

14. 4
9
21 85 6

15. 4
9
21 85 6

16. 4 921 85 6

17. 4 921 85 6
Can we speed up the find-min operations?

18. 4
9
21
8
5
6
Organize the elements so that finding minimum is “easy”.

19. 4
9
21
8
5
6
Organize the elements so that deleting the minimum
element is “easy”.

20. Store a set of n distinct numbers.
Inserting a new element and deleting the minimum
element can be done in time proportional to (log n).
GOAL

21. All jumbled up
4 9 2 1 8

22. All jumbled up
4 9 2 1 8 3

23. All jumbled up
4 9 2 1 8 3 6

24. All jumbled up
4 9 2 1 8 3
Insert: O(1)
6

25. All jumbled up
4 9 2 1 8 3
Insert: O(1) Delete-Min: O(n)
6

26. All sorted
9 8 6 3 1

27. All sorted
9 8 6 3

28. All sorted
9 8 6 3
Delete-Min: O(1)

29. All sorted
9 8 6 3
Delete-Min: O(1)
4

30. All sorted
9 8 6 3
Delete-Min: O(1)
47

31. All sorted
9 8 6 3
Insert: O(n) Delete-Min: O(1)
47

32. In-between?
Invariant: The first element of
the array is the minimum.

33. In-between?
Invariant: The first element of
the array is the minimum.

34. In-between?
Invariant: The first element of
the array is the minimum.

35. In-between?
Invariant: The first element of
the array is the minimum.

36. In-between?
Invariant: The first element of
the array is the minimum.

37. In-between?
Invariant: The first element of
the array is the minimum.

38. In-between?
Invariant: The first element of
the array is the minimum.

39. In-between?
Invariant: The first element of
the array is the minimum.

40. 4
9
21
8
5
6

41. 6
1
59
24
8
x is a child of y: x is larger than y.

42. 6
1
59
24
8

43. 6
1
59
24
8

44. 6
1
59
24
8

45. 61 5924 8

46. 61 5924 8
6
1
59
24
8

47. 61 5924 8
6
1
59
24
8
1
2 3
4 5 6 7
1 2 3 4 5 6 7

48. 61 5924 8
6
1
59
24
8
1
2 3
4 5 6 7
1 2 3 4 5 6 7
left-child(i) = 2i

49. 61 5924 8
6
1
4
1
2
4
1 2 3 4 5 6 7

50. 61 5924 8
6
1
4
1
2
4
1 2 3 4 5 6 7
left-child(i) = 2i

51. left-child(i) = 2i

52. left-child(i) = 2i
i
2i

53. left-child(i) = 2i
i+k
2i + 2k
i
2i

54. 6
1
59
24
8
Delete Min

55. 6 1
5
9
24
8
Delete Min

56. 6 1
5
9
24
8
Delete Min

57. 6 1
5
9
24
8
Delete Min

58. 6 1
5
9
24
8
Delete Min

59. 6 1
5
9
24
8
Delete Min

60. 6 1
5
9
2
4
8
Delete Min

61. 6 1
2
9
54
8
Delete Min

62. 6
2
9
54
8
Insert

63. 6
3
2
9
54
8
Insert

64. 6 3
2
9
54
8
Insert

65. 6 3
2
9
54
8
Insert

66. 6
3
2
9 5
4
8
Insert

67. 6
3
2
9 5
4
8
Insert

68. Heapify