1. Lecture No 5A Advance Analysis of Institute of Southern Punjab Multan
Department of Computer Science

2. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
2.78
7.42
0.56
1.12
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 0: step 0.

3. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
2.78
7.42
0.56
1.12
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 1: step 0.

4. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
2.78
7.42
0.56
7.42
0.56
1.12
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 2: step 0.

5. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
2.78
2.78
0.56
7.42
1.12
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 2: step 1.

6. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
2.78
7.42
1.12
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 2: step 2.

7. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
2.78
1.12
7.42
7.42
1.12
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 3: step 0.

8. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
2.78
1.12
2.78
1.12
7.42
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 3: step 1.

9. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
1.12
2.78
7.42
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 3: step 2.

10. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
1.12
2.78
7.42
1.17
7.42
1.17
0.32
6.21
4.42
3.14
7.71
Iteration 4: step 0.

11. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
1.12
1.17
2.78
2.78
1.17
7.42
0.32
6.21
4.42
3.14
7.71
Iteration 4: step 1.

12. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
1.12
1.17
2.78
7.42
0.32
6.21
4.42
3.14
7.71
Iteration 4: step 2.

13. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
1.12
1.17
2.78
7.42
0.32
7.42
0.32
6.21
4.42
3.14
7.71
Iteration 5: step 0.

14. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
1.12
1.17
0.32
2.78
2.78
0.32
7.42
6.21
4.42
3.14
7.71
Iteration 5: step 1.

15. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
1.12
0.32
1.17
1.17
0.32
2.78
7.42
6.21
4.42
3.14
7.71
Iteration 5: step 2.

16. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.56
0.32
1.12
1.12
0.32
1.17
2.78
7.42
6.21
4.42
3.14
7.71
Iteration 5: step 3.

17. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
0.56
0.32
1.12
1.17
2.78
7.42
6.21
4.42
3.14
7.71
Iteration 5: step 4.

18. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
7.42
6.21
4.42
3.14
7.71
Iteration 5: step 5.

19. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
6.21
7.42
7.42
6.21
4.42
3.14
7.71
Iteration 6: step 0.

20. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
6.21
7.42
4.42
3.14
7.71
Iteration 6: step 1.

21. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
6.21
7.42
4.42
7.42
4.42
3.14
7.71
Iteration 7: step 0.

22. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
4.42
6.21
6.21
4.42
7.42
3.14
7.71
Iteration 7: step 1.

23. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
4.42
6.21
7.42
3.14
7.71
Iteration 7: step 2.

24. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
4.42
6.21
7.42
3.14
7.42
3.14
7.71
Iteration 8: step 0.

25. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
4.42
6.21
3.14
6.21
3.14
7.42
7.71
Iteration 8: step 1.

26. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
3.14
4.42
4.42
3.14
6.21
7.42
7.71
Iteration 8: step 2.

27. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
3.14
4.42
6.21
7.42
7.71
Iteration 8: step 3.

28. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
3.14
4.42
6.21
7.42
7.71
Iteration 9: step 0.

29. 5/22/2019
Insertion Sort
Iteration i. Repeatedly swap element i with the one to its left if smaller.
Property. After ith iteration, a[0] through a[i] contain first i+1 elements in ascending order. 2 3 4 5 1 8 9
Array index 6 7
Value
0.32
0.56
1.12
1.17
2.78
3.14
4.42
6.21
7.42
7.71
Iteration 10: DONE.

30. 5 2 4 6 1 3 Insertion Sort input array
at each iteration, the array is divided in two sub-arrays:
left sub-array
right sub-array
unsorted
sorted

31. Insertion Sort

32. An Example: Insertion Sort
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

33. An Example: Insertion Sort30 10 40 20
i =  j =  key =  A[j] =  A[j+1] =  1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

34. An Example: Insertion Sort30 10 40 20
i = 2 j = 1 key = 10 A[j] = 30 A[j+1] = 10 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

35. An Example: Insertion Sort30 30 40 20
i = 2 j = 1 key = 10 A[j] = 30 A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

36. An Example: Insertion Sort30 30 40 20
i = 2 j = 1 key = 10 A[j] = 30 A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

37. An Example: Insertion Sort30 30 40 20
i = 2 j = 0 key = 10 A[j] =  A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

38. An Example: Insertion Sort30 30 40 20
i = 2 j = 0 key = 10 A[j] =  A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

39. An Example: Insertion Sort10 30 40 20
i = 2 j = 0 key = 10 A[j] =  A[j+1] = 10 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

40. An Example: Insertion Sort10 30 40 20
i = 3 j = 0 key = 10 A[j] =  A[j+1] = 10 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

41. An Example: Insertion Sort10 30 40 20
i = 3 j = 0 key = 40 A[j] =  A[j+1] = 10 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

42. An Example: Insertion Sort10 30 40 20
i = 3 j = 0 key = 40 A[j] =  A[j+1] = 10 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

43. An Example: Insertion Sort10 30 40 20
i = 3 j = 2 key = 40 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

44. An Example: Insertion Sort10 30 40 20
i = 3 j = 2 key = 40 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

45. An Example: Insertion Sort10 30 40 20
i = 3 j = 2 key = 40 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

46. An Example: Insertion Sort10 30 40 20
i = 4 j = 2 key = 40 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

47. An Example: Insertion Sort10 30 40 20
i = 4 j = 2 key = 20 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

48. An Example: Insertion Sort10 30 40 20
i = 4 j = 2 key = 20 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

49. An Example: Insertion Sort10 30 40 20
i = 4 j = 3 key = 20 A[j] = 40 A[j+1] = 20 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

50. An Example: Insertion Sort10 30 40 20
i = 4 j = 3 key = 20 A[j] = 40 A[j+1] = 20 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

51. An Example: Insertion Sort10 30 40 40
i = 4 j = 3 key = 20 A[j] = 40 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

52. An Example: Insertion Sort10 30 40 40
i = 4 j = 3 key = 20 A[j] = 40 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

53. An Example: Insertion Sort10 30 40 40
i = 4 j = 3 key = 20 A[j] = 40 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

54. An Example: Insertion Sort10 30 40 40
i = 4 j = 2 key = 20 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

55. An Example: Insertion Sort10 30 40 40
i = 4 j = 2 key = 20 A[j] = 30 A[j+1] = 40 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

56. An Example: Insertion Sort10 30 30 40
i = 4 j = 2 key = 20 A[j] = 30 A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

57. An Example: Insertion Sort10 30 30 40
i = 4 j = 2 key = 20 A[j] = 30 A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

58. An Example: Insertion Sort10 30 30 40
i = 4 j = 1 key = 20 A[j] = 10 A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

59. An Example: Insertion Sort10 30 30 40
i = 4 j = 1 key = 20 A[j] = 10 A[j+1] = 30 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

60. An Example: Insertion Sort10 20 30 40
i = 4 j = 1 key = 20 A[j] = 10 A[j+1] = 20 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
David Luebke /22/2019

61. An Example: Insertion Sort10 20 30 40
i = 4 j = 1 key = 20 A[j] = 10 A[j+1] = 20 1 2 3 4
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
Done!
David Luebke /22/2019

62. What is the precondition for this loop?
Insertion Sort
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
What is the precondition
for this loop?
David Luebke /22/2019

63. How many times will this loop execute?
Insertion Sort
InsertionSort(A, n) { for i = 2 to n { key = A[i] j = i - 1; while (j > 0) and (A[j] > key) { A[j+1] = A[j] j = j - 1 } A[j+1] = key } }
How many times will this loop execute?
David Luebke /22/2019

64. Statement Effort Insertion Sort InsertionSort(A, n) {
for i = 2 to n { c1n
key = A[i] c2(n-1)
j = i - 1; c3(n-1)
while (j > 0) and (A[j] > key) { c4T
A[j+1] = A[j] c5(T-(n-1))
j = j c6(T-(n-1)) }
A[j+1] = key c7(n-1) } }
T = t2 + t3 + … + tn where ti is number of while expression evaluations for the ith for loop iteration
David Luebke /22/2019