1. The Power of Computing Algorithms
Daryle J. Serrant – Past Undergraduate Student
Debasis Mitra –
Florida Institute of Technology
Computer Science

2. What is an Algorithm? Cooking <- Recipes
Building construction <- Blue print
Software <- Algorithm

3. Example Problem 1 Best Return on a Stock
We have daily stock prices over a period:
Gains as positive
Loss as negative
When should we have bought and when should we have sold

4. Maximum Customers visited in a Store
Problem Example 2
Maximum Customers visited in a Store
# Customers visited & left every hour is tracked
Incoming number as positive
Leaving number as negative
Between which hours maximum #customers were in the store

5. A Computing Problem: Maximum Subsequence
What is common between the two examples?
Can you find the maximum sub-sequence
of the following numbers:
3, 4, -8, 1, 9, -2, 3, -1
Answer: {1, 9, -2, 3} for the max sum 11.
Question: How fast can we get the answer?
Answer: Depends on the ALGORITHM!

6. The Workaholic Approach
Go over every possible subsequence:
Day 1 – Day 2, Day 1 – Day 3, etc.
Find the subsequence with the largest sum.
Image Source:

7. Day 1 – Day 1 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 67
Day 1 – Day 1
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: = 3
Max Sum: 3
Addition Counter: 1
Maximum Subsequence: Day 1

8. Day 1 – Day 2 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Start
End
Sum: 0 + 3
Addition Counter: 2

9. Day 1 – Day 2 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Start
End
Sum: = 7
Max Sum: 3
Addition Counter: 3
Maximum Subsequence: Day 1
Max Sum is now 7 and Maximum Subsequence is now Day 1 - 2

10. Day 1 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1010
Day 1 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 3
Addition Counter: 4

11. Day 1 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1111
Day 1 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 5

12. Day 1 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1212
Day 1 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: = -1
Max Sum: 7
Addition Counter: 6
Maximum Subsequence: Day 1 - 2
No change

13. Day 1 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1313
Day 1 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 3
Addition Counter: 7

14. Day 1 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1414
Day 1 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 8

15. Day 1 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1515
Day 1 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 9

16. Day 1 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1616
Day 1 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – = 0
Max Sum: 7
Addition Counter: 10
Maximum Subsequence: Day 1 - 2
No changes

17. Day 1 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1717
Day 1 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 3
Addition Counter: 11

18. Day 1 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1818
Day 1 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 12

19. Day 1 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
1919
Day 1 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 13

20. Day 1 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2020
Day 1 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 14

21. Day 1 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2121
Day 1 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – = 9
Max Sum: 7
Addition Counter: 15
Maximum Subsequence: Day 1 - 2
Max Sum is now 9 and Maximum Subsequence is Day 1 – Day 5

22. Day 1 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2222
Day 1 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 3
Addition Counter: 16

23. Day 1 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2323
Day 1 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 17

24. Day 1 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2424
Day 1 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 18

25. Day 1 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2525
Day 1 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 19

26. Day 1 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2626
Day 1 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: –
Addition Counter: 20

27. Day 1 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2727
Day 1 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2 = 7
Max Sum: 9
Addition Counter: 21
Maximum Subsequence: Day 1 - 5
No change

28. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2828
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 3
Addition Counter: 22

29. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
2929
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 23

30. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3030
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 24

31. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3131
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 25

32. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3232
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: –
Addition Counter: 26

33. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3333
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2
Addition Counter: 27

34. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3434
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – = 10
Max Sum: 9
Addition Counter: 28
Maximum Subsequence: Day 1 - 5
Max Sum is now 10 and Maximum Subsequence is Day 1 - 7

35. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3535
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 3
Addition Counter: 29

36. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3636
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 30

37. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3737
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 31

38. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3838
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 32

39. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
3939
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: –
Addition Counter: 33

40. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4040
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2
Addition Counter: 34

41. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4141
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2 + 3
Addition Counter: 35

42. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4242
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – – 1 = 9
Max Sum: 10
Addition Counter: 36
Maximum Subsequence: Day 1 - 7
No changes

43. Day 2 – Day 2 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4343
Day 2 – Day 2
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: = 4
Max Sum: 10
Addition Counter: 37
Maximum Subsequence: Day 1 - 7
No changes

44. Day 2 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4444
Day 2 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 4
Addition Counter: 38

45. Day 2 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4545
Day 2 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 = -4
Max Sum: 10
Addition Counter: 39
Maximum Subsequence: Day 1 - 7
No changes

46. Day 2 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4646
Day 2 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 4
Addition Counter: 40

47. Day 2 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4747
Day 2 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 41

48. Day 2 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4848
Day 2 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – = -3
Max Sum: 10
Addition Counter: 42
Maximum Subsequence: Day 1 - 7
No changes

49. Day 2 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
4949
Day 2 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 4
Addition Counter: 43

50. Day 2 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5050
Day 2 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 44

51. Day 2 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5151
Day 2 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 45

52. Day 2 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5252
Day 2 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – = 6
Max Sum: 10
Addition Counter: 46
Maximum Subsequence: Day 1 - 7
No changes

53. Day 2 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5353
Day 2 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 4
Addition Counter: 47

54. Day 2 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5454
Day 2 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 48

55. Day 2 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5555
Day 2 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 49

56. Day 2 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5656
Day 2 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: –
Addition Counter: 50

57. Day 2 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5757
Day 2 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2 = 4
Max Sum: 10
Addition Counter: 51
Maximum Subsequence: Day 1 - 7
No changes

58. Day 2 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5858
Day 2 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 4
Addition Counter: 52

59. Day 2 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
5959
Day 2 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 53

60. Day 2 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6060
Day 2 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 54

61. Day 2 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6161
Day 2 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: –
Addition Counter: 55

62. Day 2 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6262
Day 2 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2
Addition Counter: 56

63. Day 2 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6363
Day 2 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – = 7
Max Sum: 10
Addition Counter: 57
Maximum Subsequence: Day 1 - 7
No changes

64. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6464
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 4
Addition Counter: 58

65. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6565
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8
Addition Counter: 59

66. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6666
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 8 + 1
Addition Counter: 60

67. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6767
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: –
Addition Counter: 61

68. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6868
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2
Addition Counter: 62

69. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
6969
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 2 + 3
Addition Counter: 63

70. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7070
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – – 1 = 6
Max Sum: 10
Addition Counter: 64
Maximum Subsequence: Day 1 - 7
No changes

71. Day 3 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7171
Day 3 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8 = -8
Max Sum: 10
Addition Counter: 65
Maximum Subsequence: Day 1 - 7
No changes

72. Day 3 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7272
Day 3 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8 = -8
Addition Counter: 66

73. Day 3 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7373
Day 3 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – = -7
Max Sum: 10
Addition Counter: 67
Maximum Subsequence: Day 1 - 7
No changes

74. Day 3 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7474
Day 3 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8 = -8
Addition Counter: 68

75. Day 3 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7575
Day 3 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8 + 1
Addition Counter: 69

76. Day 3 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7676
Day 3 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – = 2
Max Sum: 10
Addition Counter: 70
Maximum Subsequence: Day 1 - 7
No changes

77. Day 3 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7777
Day 3 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8
Addition Counter: 71

78. Day 3 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7878
Day 3 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8 + 1
Addition Counter: 72

79. Day 3 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
7979
Day 3 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 –
Addition Counter: 73

80. Day 3 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8080
Day 3 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – – 2 = 0
Max Sum: 10
Addition Counter: 74
Maximum Subsequence: Day 1 - 7
No changes

81. Day 3 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8181
Day 3 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8
Addition Counter: 75

82. Day 3 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8282
Day 3 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8 + 1
Addition Counter: 76

83. Day 3 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8383
Day 3 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 –
Addition Counter: 77

84. Day 3 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8484
Day 3 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – – 2
Addition Counter: 78

85. Day 3 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8585
Day 3 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – – = 3
Max Sum: 10
Addition Counter: 79
Maximum Subsequence: Day 1 - 7
No changes

86. Day 3 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8686
Day 3 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8
Addition Counter: 80

87. Day 3 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8787
Day 3 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 8 + 1
Addition Counter: 81

88. Day 3 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8888
Day 3 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 –
Addition Counter: 82

89. Day 3 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
8989
Day 3 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – – 2
Addition Counter: 83

90. Day 3 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9090
Day 3 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – – 2 + 3
Addition Counter: 84

91. Day 3 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9191
Day 3 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – – – 1 = 2
Max Sum: 10
Addition Counter: 85
Maximum Subsequence: Day 1 - 7
No changes

92. Day 4 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9292
Day 4 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: = 1
Max Sum: 10
Addition Counter: 86
Maximum Subsequence: Day 1 - 7
No changes

93. Day 4 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9393
Day 4 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 1
Addition Counter: 87

94. Day 4 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9494
Day 4 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: = 10
Max Sum: 10
Addition Counter: 88
Maximum Subsequence: Day 1 - 7
No changes

95. Day 4 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9595
Day 4 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 1
Addition Counter: 89

96. Day 4 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9696
Day 4 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 90

97. Day 4 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9797
Day 4 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2 = 8
Max Sum: 10
Addition Counter: 91
Maximum Subsequence: Day 1 - 7
No changes

98. Day 4 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9898
Day 4 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 1
Addition Counter: 92

99. Day 4 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
9999
Day 4 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 93

100. Day 4 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2
Addition Counter: 94

101. Day 4 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – = 11
Max Sum: 10
Addition Counter: 95
Maximum Subsequence: Day 1 - 7
Max Sum is 11 and Maximum Subsequence is now Day 4 – Day 7

102. Day 4 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 1
Addition Counter: 96

103. Day 4 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum:
Addition Counter: 97

104. Day 4 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2
Addition Counter: 98

105. Day 4 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2 + 3
Addition Counter: 99

106. Day 4 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 1 = 10
Max Sum: 11
Addition Counter:
100
Maximum Subsequence: Day 4 - 7
No changes

107. Day 5 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: = 9
Max Sum: 11
Addition Counter:
101
Maximum Subsequence: Day 4 - 7
No changes

108. Day 5 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 9
Addition Counter:
102

109. Day 5 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2 = 7
Max Sum: 11
Addition Counter:
103
Maximum Subsequence: Day 4 - 7
No changes

110. Day 5 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 9
Addition Counter:
104

111. Day 5 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2
Addition Counter:
105

112. Day 5 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – = 10
Max Sum: 11
Addition Counter:
106
Maximum Subsequence: Day 4 - 7
No changes

113. Day 5 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 9
Addition Counter:
107

114. Day 5 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2
Addition Counter:
108

115. Day 5 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 2 + 3
Addition Counter:
109

116. Day 5 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – – 1 = 9
Max Sum: 11
Addition Counter:
110
Maximum Subsequence: Day 4 - 7
No changes

117. Day 6 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 2 = -2
Max Sum: 11
Addition Counter:
111
Maximum Subsequence: Day 1 - 7
No changes

118. Day 6 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 2
Addition Counter:
112

119. Day 6 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – = 1
Max Sum: 11
Addition Counter:
113
Maximum Subsequence: Day 4 - 7
No changes

120. Day 6 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 2
Addition Counter:
114

121. Day 6 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 2 + 3
Addition Counter:
115

122. Day 6 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – – 1 = 0
Max Sum: 11
Addition Counter:
116
Maximum Subsequence: Day 4 - 7
No changes

123. Day 7 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 7 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: = 3
Max Sum: 11
Addition Counter:
117
Maximum Subsequence: Day 4 - 7
No changes

124. Day 7 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 7 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 + 3
Addition Counter:
118

125. Day 7 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 7 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: – 1 = 2
Max Sum: 11
Addition Counter:
119
Maximum Subsequence: Day 4 - 7
No changes

126. Day 8 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 8 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Start
End
Sum: 0 – 1 = -1
Max Sum: 11
Addition Counter:
120
Maximum Subsequence: Day 4 - 7
No changes

127. Analysis
Each sub-sequence between green & red ptrs is aggregated, at most 8 addition counting operations per sub-sequence,
… by using the black pointer
There is 32 subsequences in total
That is a lot of adding to do, don’t you think?
Imagine having to find the maximum sub-sequence over100 days. Or, maybe 20,000 days.
Given n days of stock data, the number of addition operations that you will perform using the workaholic approach will be in the ballpark of around n3
Computers have to do exactly the same computation!!

128. Improve!
Do we really need to run the black/middle pointer over and over again on same numbers?
As the red/right pointer moves cann’t we just add that number to the previous sum?
Computers have to do exactly the same computation!!

129. The Conservative Approach
The Workaholic approach requires a lot of addition operations
So much, that it’s impractical to use for large data sets
We can save a lot of time by using accumulated sums from the previous sub-sequences.
Let us call this the Conservative approach.
Photo source:

130. Day 1 – Day 1 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Task: black triangle on red??
Day 1 – Day 1
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 3
Max Sum: 3
Maximum Subsequence: Day 1 – Day 1
Addition Counter: 1

131. Day 1 – Day 2 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 1 – Day 2
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 7
Max Sum: 3
Maximum Subsequence: Day 1 – Day 1
Addition Counter: 2
Max Sum is now 7 and Maximum Subsequence is Day 1 – Day 2

132. Day 1 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 1 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 7 – 8 = -1
Max Sum: 7
Maximum Subsequence: Day 1 – Day 2
Addition Counter: 3
No changes

133. Day 1 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 1 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 0
Max Sum: 7
Maximum Subsequence: Day 1 – Day 2
Addition Counter: 4
No changes

134. Day 1 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 1 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 9
Max Sum: 7
Maximum Subsequence: Day 1 – Day 2
Addition Counter: 5
Max Sum is now 9 and Maximum Subsequence is Day 1 – Day 5

135. Day 1 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 1 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 9 – 2 = 7
Max Sum: 9
Maximum Subsequence: Day 1 – Day 5
Addition Counter: 6
No changes

136. Day 1 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 1 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 10
Max Sum: 9
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 7
Max Sum is now 10 and Maximum Subsequence is Day 1 – Day 7

137. Day 1 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 1 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 10 – 1 = 9
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 8
No changes

138. Day 2 – Day 2 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 2 – Day 2
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 4
Max Sum: 11
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 9
No changes

139. Day 2 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 2 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 4 – 8 = -4
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 10
No changes

140. Day 2 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 2 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = -3
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 11
No changes

141. Day 2 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 2 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 6
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 12
No changes

142. Day 2 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 2 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 6 – 2 = 4
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 13
No changes

143. Day 2 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 2 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 7
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 14
No changes

144. Day 2 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 2 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 7 – 1 = 6
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 15
No changes

145. Day 3 – Day 3 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 3 – Day 3
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 0 – 8 = 8
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 16
No changes

146. Day 3 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 3 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = -7
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 17
No changes

147. Day 3 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 3 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 2
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 18
No changes

148. Day 3 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 3 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 2 – 2 = 0
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 19
No changes

149. Day 3 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 3 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 3
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 20
No changes

150. Day 3 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 3 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 3 – 1 = 2
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 21
No changes

151. Day 4 – Day 4 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 4
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 1
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 22
No changes

152. Day 4 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 10
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 23
No changes

153. Day 4 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 10 – 2 = 8
Max Sum: 10
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 24
No changes

154. Day 4 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 11
Max Sum: 11
Maximum Subsequence: Day 1 – Day 7
Addition Counter: 25
Max Sum is 11 and Maximum Subsequence is now Day 4 – Day 7

155. Day 4 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 4 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 11 – 1 = 10
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 26
No changes

156. Day 5 – Day 5 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
No third black pointer any more
Day 5 – Day 5
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 9
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 27
No changes

157. Day 5 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 9 – 2 = 7
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 28
No changes

158. Day 5 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 10
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 29
No changes

159. Day 5 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 5 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 10 – 1 = 9
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 30
No changes

160. Day 6 – Day 6 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 6
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 0 – 2 = -2
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 31
No changes

161. Day 6 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 1
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 32
No changes

162. Day 6 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 6 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 1 – 1 = 0
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 33
No changes

163. Day 7 – Day 7 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 7 – Day 7
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: = 3
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 34
No changes

164. Day 7 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 7 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 3 – 1 = 2
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 35
No changes

165. Day 8 – Day 8 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6
Day 8 – Day 8
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum: 0 – 1 = -1
Max Sum: 11
Maximum Subsequence: Day 4 – Day 7
Addition Counter: 36
No changes

166. Some more Analysis
We are now only required to do 1 additional counting operation for each sub-sequence compared to at most 8 in the workaholic approach
For larger data sets, the number of addition operations will be in the ball park of around n2, where n is the size of the set.
Workaholic approach had this as n3

167. The Butcher’s Approach
The Conservative approach reduced the number of additions we needed to do by a factor of n.
This is good, but we can do even better than this.
Instead of solving the entire problem as a whole, how about cutting the problem down into smaller pieces that we can solve that doesn’t require too much addition, solve those problems, and combine the results to get the final result.
Let us call this approach the Butcher’s approach.
Photo source:

168. Combine results from smaller problems
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1

169. Solve Max Subsequence for Day 1 – Day 43 4 -8 1 9 -2 3 -1
Left Max Sum (Day 1 – Day 4) = 7
Left Max Subsequence = Day 1 – Day 2
Addition Counter: 9

170. Solve Max Subsequence for Day 5 – Day 83 4 -8 1 9 -2 3 -1
Left Max Sum (Day 1 – Day 4) = 7
Right Max Sum (Day 5 – Day 8) = 10
Left Max Subsequence = Day – Day 2
Right Max Subsequence = Day 5 – Day 7
Addition Counter: 18

171. Which side has the Largest Sum?
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Sum (Day 1 – Day 4) = 7
Right Max Sum (Day 5 – Day 8) = 10
Left Max Subsequence = Day – Day 2
Right Max Subsequence = Day 5 – Day 7
Left Max Sum > Right Max Sum? No
Addition Counter: 18

172. Which side has the Largest Sum?
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Sum (Day 1 – Day 4) = 7
Right Max Sum (Day 5 – Day 8) = 10
Left Max Subsequence = Day – Day 2
Right Max Subsequence = Day 5 – Day 7
Left Max Sum > Right Max Sum? No
Max Side Subsequence = Day 5 – Day 7
Addition Counter: 18

173. Find the Middle Max Subsequence
Find the starting point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum = 1
Left Max Border Sum = 0
Mid Start = Day 4
Left Max Border Sum is now 1
Addition Counter: 18

174. Find the Middle Max Subsequence
Find the starting point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum = 1 – 8 = -7
Left Max Border Sum = 1
Mid Start = Day 4
No changes made to Left Max Border Sum
Addition Counter: 19

175. Find the Middle Max Subsequence
Find the starting point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum = = -3
Left Max Border Sum = 1
Mid Start = Day 4
No changes made to Left Max Border Sum
Addition Counter: 20

176. Find the Middle Max Subsequence
Find the starting point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Sum = = 0
Left Max Border Sum = 1
Mid Start = Day 4
No changes made to Left Max Border Sum
Addition Counter: 21

177. Find the Middle Max Subsequence
Find the ending point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Border Sum = 1
Sum = 9
Mid Start = 4
Right Max Border Sum = 0
Mid End= Day 5
Right Max Border Sum is now 9
Addition Counter: 22

178. Find the Middle Max Subsequence
Find the ending point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Border Sum = 1
Sum = 9 – 2 = 7
Mid Start = 4
Right Max Border Sum = 9
Mid End= Day 5
No changes made to Right Max Border Sum
Addition Counter: 23

179. Find the Middle Max Subsequence
Find the ending point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Border Sum = 1
Sum = = 10
Mid Start = 4
Right Max Border Sum = 9
Mid End= Day 5
Right Max Border Sum is now 10 and Mid End is Day 7
Addition Counter: 24

180. Find the Middle Max Subsequence
Find the ending point
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Border Sum = 1
Sum = 10 – 1 = 9
Mid Start = 4
Right Max Border Sum = 10
Mid End= Day 7
No changes made to Right Max Border Sum
Addition Counter: 25

181. Mid Max Sum = Left Max Border Sum + Right Max Border Sum = 11
Finding Middle Max Subsequence
Compute Middle Max Sum
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Border Sum = 1
Right Max Border Sum = 10
Mid Start = 4
Mid End= Day 7
Mid Max Sum = Left Max Border Sum + Right Max Border Sum = 11
Range: Day 4 through Day 7
Addition Counter: 25

182. Find Max Subsequence for Day 1 – Day 8
Compare subsequence sums
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Border Sum = 1
Right Max Border Sum = 10
Mid Start = 4
Mid End= Day 7
Mid Max Sum = Left Max Border Sum + Right Max Border Sum = 11
Mid Max Sum > Max Side Sum? Yes
Addition Counter: 25

183. Find Max Subsequence for Day 1 – Day 8
Max Side Subsequence = Day 5 – Day 7
Left Max Sum = 7
Right Max Sum = 10
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
Left Max Border Sum = 1
Right Max Border Sum = 10
Mid Start = 4
Mid End= Day 7
Left Max Border Sum + Right Max Border Sum = 11
Max Sum is 11
Max Subsequence is Day 4 – Day 7
Addition Counter: 25

184. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Addition Counter:

185. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Addition Counter:

186. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Day 1
Day 2 3 4
Addition Counter:

187. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Day 1
Day 2 3 4
Day 1 3
Addition Counter:

188. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Day 1
Day 2 3 4
Day 1 3
> 0 ? Yes
Addition Counter:

189. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Day 1
Day 2 3 4
Left Max Sum = 3
Addition Counter:

190. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Day 1
Day 2 3 4
Day 2
Left Max Sum = 3 4
> 0 ? Yes
Addition Counter:

191. Solving for Left Half Addition Counter: Day 1 Day 2 Day 3 Day 4 Day 5-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Day 1
Day 2 3 4
Left Max Sum = 3
Left Max Subsequence = Day 1 – Day 1
Right Max Sum = 4
Right Max Subsequence = Day 2 – Day 2
Max Side Subsequence = Day 2 – Day 2
Addition Counter:

192. Solving for Left Half Addition Counter: 1 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Max Side Subsequence = Day 2 – Day 2
Left Max Sum = 3
Right Max Sum = 4
Day 1
Day 2 3 4
Left Max Border Sum = 0 + 3
Addition Counter: 1
midStart = 1

193. Solving for Left Half Addition Counter: 2 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Max Side Subsequence = Day 2 – Day 2
Left Max Sum = 3
Right Max Sum = 4
Day 1
Day 2 3 4
Left Max Border Sum = 3
Right Max Border Sum = 0 + 4
Addition Counter: 2
midStart = 1
midStart = 2

194. Solving for Left Half Addition Counter: 2 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Max Side Subsequence = Day 2 – Day 2
Left Max Sum = 3
Right Max Sum = 4
Day 1
Day 2 3 4
Left Max Border Sum = 3
Right Max Border Sum = 0 + 4
midStart = 1
midStart = 2
Addition Counter: 2
Left Max Border Sum + Right Max Border Sum = 7
Max Sum is 7
Maximum Subsequence is Day 1 – Day 2

195. Solving for Left Half Addition Counter: 2 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Day 3
Day 4 -8 1
Addition Counter: 2

196. Solving for Left Half Addition Counter: 2 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Day 3
Day 4 -8 1
Day 3 -8
Addition Counter: 2

197. Solving for Left Half Addition Counter: 2 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Day 3
Day 4 -8 1
Day 4 1
Addition Counter: 2
> 0 ? Yes

198. Solving for Left Half Addition Counter: 2 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Day 3
Day 4 -8 1
Left Max Sum = 0
Left Max Subsequence = Day 3 – Day 3
Right Max Sum = 1
Right Max Subsequence = Day 4 – Day 4
Addition Counter: 2
Max Side Subsequence = Day 4 – Day 4

199. Solving for Left Half Addition Counter: 3 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Max Side Subsequence = Day 4 – Day 4
Left Max Sum = 0
Right Max Sum = 1
Day 3
Day 4 -8 1
Addition Counter: 3
Left Max Border Sum = 0
Mid Start = Day 3

200. Solving for Left Half Addition Counter: 4 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Max Side Subsequence = Day 4 – Day 4
Left Max Sum = 0
Right Max Sum = 1
Day 3
Day 4 -8 1
Addition Counter: 4
Right Max Border Sum = 1
Left Max Border Sum = 0
Mid Start = Day 3
Mid End = Day 4

201. Solving for Left Half Addition Counter: 5 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Max Side Subsequence = Day 4 – Day 4
Left Max Sum = 0
Right Max Sum = 1
Day 3
Day 4 -8 1
Addition Counter: 5
Left Max Border Sum = 0
Right Max Border Sum = 1
Mid Start = Day 3
Mid End = Day 4
Max Sum is 1
Maximum Subsequence = Day 3 – Day 4

202. Solving for Left Half Addition Counter: 5 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Sum = 7
Left Max Subsequence = Day 1 – Day 2
Right Max Sum = 1
Right Max Subsequence = Day 3 – Day 4
Max Side Subsequence = Day 1 – Day 2
Addition Counter: 5

203. Solving for Left Half Addition Counter: 6 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Max Side Subsequence = Day 1 – Day 2
Left Max Sum = 7
Right Max Sum = 1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Sum = 4
Left Max Border Sum = 0
Mid Start = Day 2
Left Max Border Sum is now 4
Addition Counter: 6

204. Solving for Left Half Addition Counter: 7 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Max Side Subsequence = Day 1 – Day 2
Left Max Sum = 7
Right Max Sum = 1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Sum = = 7
Left Max Border Sum = 4
Mid Start = Day 2
Left Max Border Sum is now 7 and Mid Start is now Day 1
Addition Counter: 7

205. Solving for Left Half Addition Counter: 8 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Max Side Subsequence = Day 1 – Day 2
Left Max Sum = 7
Right Max Sum = 1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Border Sum = 7
Sum = -8
Mid Start = Day 1
Mid End = Day 3
Right Max Border Sum = 0
No change to Right Max Border Sum
Addition Counter: 8

206. Solving for Left Half Addition Counter: 9 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Max Side Subsequence = Day 1 – Day 2
Left Max Sum = 7
Right Max Sum = 1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Border Sum = 7
Sum = = --7
Right Max Border Sum = 0
Mid Start = Day 1
Mid End = Day 3
No change to Right Max Border Sum or Mid End
Addition Counter: 9

207. Solving for Left Half Addition Counter: 9 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Max Side Subsequence = Day 1 – Day 2
Left Max Sum = 7
Right Max Sum = 1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Border Sum = 7
Right Max Border Sum = 0
Mid Start = Day 1
Mid End = Day 3
Addition Counter: 9

208. Solving for Left Half Addition Counter: 9 Day 1 Day 2 Day 3 Day 4-8 1 9 -2 3 -1
Max Side Subsequence = Day 1 – Day 2
Left Max Sum = 7
Right Max Sum = 1
Day 1
Day 2
Day 3
Day 4 3 4 -8 1
Left Max Border Sum = 7
Right Max Border Sum = 0
Mid Start = Day 1
Mid End = Day 3
Left Max Border Sum + Right Max Border Sum = 7
Addition Counter: 9
Max Sum is 7
Max Subsequence is Day 1 – Day 2

209. Analysis of Butcher’s Approach
The amount of addition needed for the Butcher’s approach compared to the Workaholic and Conservative approach is extremely small:
The simplest maximum subsequence problem is one involving one day. These small problems require no addition.
The amount of addition needed from smaller sub problems to slightly larger sub problems grows slowly because we’re doing fewer re-calculations of the same numbers than the last two approaches.
We’re simply combining results of the smaller problems.
For larger sized problems, the amount of addition that needs to be done is around the ball park of n*log2(n)

210. Fourth Algorithm: The Smart A$$ Approach
We can do even better than the Butcher’s method by making a key observation about the maximum sub-sequence problem – we will NEVER want a negative stock sum!
If that’s the case, then we can just ignore additions that become negative when we solve the problem.
I call this approach the Smart A$$ approach!
This algorithm is very similar to the Conservative approach. However unlike conservative approach, we ignore subsequences with negative accumulated sums.
Photo source:

211. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cstart
cend
Sum: 3
Sum < 0? No
Addition Counter: 1
Max Sum: 3
Maximum Subsequence: Day 1 – Day 1

212. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cstart
cend
Sum: = 7
Sum < 0? No
Max Sum was: 3
Addition Counter: 2
Maximum Subsequence was: Day 1 – Day 1
Max Sum is now 7 and Maximum Subsequence is Day 1 – Day 2

213. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cend
cstart
Sum: 7 – 8 = -1
Sum < 0? Yes
Max Sum: 7
Addition Counter: 3
Maximum Subsequence: Day 1 – Day 2
Sum is now 0

214. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cend
cstart
Sum: = 1
Sum < 0? No
Max Sum: 7
Addition Counter: 4
Maximum Subsequence: Day 1 – Day 2
No changes

215. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cstart
cend
Sum: = 10
Sum < 0? No
Max Sum: 7
Addition Counter: 5
Maximum Subsequence: Day 1 – Day 2
Max Sum is now 10 and Maximum Subsequence is Day 4 – Day 5

216. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cstart
cend
Sum: 10 – 2 = 8
Sum < 0? No
Max Sum: 10
Addition Counter: 6
Maximum Subsequence: Day 4 – Day 5
No changes

217. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cstart
cend
Sum: = 11
Sum < 0? No
Max Sum: 10
Addition Counter: 7
Maximum Subsequence: Day 4 – Day 5
Max Sum is 11 and Max Subsequence is Day 4 – Day 7

218. 3 4 -8 1 9 -2 3 -1 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8
Task: remove black triangle
mstart
mend
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8 3 4 -8 1 9 -2 3 -1
cstart
cend
Sum:11 – 1 = 10
Sum < 0? No
Max Sum: 11
Addition Counter: 8
Maximum Subsequence: Day 4 – Day 7
No changes

219. Last But Not Least – The Analysis
The total number of addition operations needed for the Smart A$$ approach is… Amazing isn’t it!
Workaholic = 120, Conservative = 36, Butcher’s = 25
For 1000 days:
Workaholic = 167,167,000
Conservative = 500,500
Butcher’s = 9,976
SmartA$$ = 1,000
The Smart A$$ approach is the most efficient algorithm
The amount of addition required is around the size of the data set n.

220. Thanks,
but!
Daryle J. Serrant
Debasis Mitra

221. Your feedback will be much appreciated.
1. Is this talk useful to you?
2. If yes, can you describe, how?
3. If no, can you provide any suggestion on what we needed to do to explain algorithms more?
5. Any suggestion on where else may we present this?
6. Do you have any question for us?
Debasis Mitra, Computer Science, FIT –
Daryle J. Serrant –