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[object Object],[object Object],[object Object]
Review: Dynamic programming ,[object Object],[object Object],[object Object],[object Object],[object Object]
Properties of a problem that can be solved with dynamic programming ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Review: Longest Common Subsequence (LCS) ,[object Object],[object Object],[object Object],[object Object]
Review: Longest Common Subsequence (LCS) continued ,[object Object],[object Object],[object Object],[object Object],c[m,n] is the final solution
Review: Longest Common Subsequence (LCS) ,[object Object],[object Object]
0-1 Knapsack problem ,[object Object],[object Object],[object Object]
0-1 Knapsack problem: a picture w i b i 10 9 8 5 5 4 4 3 3 2 Weight Benefit value This is a knapsack Max weight: W = 20 Items W = 20
0-1 Knapsack problem ,[object Object],[object Object],[object Object]
0-1 Knapsack problem: brute-force approach ,[object Object],[object Object],[object Object],[object Object]
0-1 Knapsack problem: brute-force approach ,[object Object],[object Object],[object Object],Let’s try this: If items are labeled 1..n , then a subproblem would be to find an optimal solution for S k = {items labeled 1, 2, .. k}
Defining a Subproblem ,[object Object],[object Object],[object Object],[object Object]
Defining a Subproblem Max weight: W = 20 For S 4 : Total weight: 14; total benefit: 20 w i b i 10 8 5 5 4 4 3 3 2 Weight Benefit 9 Item # 4 3 2 1 5 S 4 S 5 For S 5 : Total weight: 20 total benefit: 26 Solution for S 4 is not part of the solution for S 5 !!! ? w 1 =2 b 1 =3 w 2 =4 b 2 =5 w 3 =5 b 3 =8 w 4 =3 b 4 =4 w 1 =2 b 1 =3 w 2 =4 b 2 =5 w 3 =5 b 3 =8 w 4 =9 b 4 =10
Defining a Subproblem (continued) ,[object Object],[object Object],[object Object],[object Object]
Recursive Formula for subproblems ,[object Object],[object Object],[object Object],[object Object]
Recursive Formula ,[object Object],[object Object],[object Object]
0-1 Knapsack Algorithm ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Running time ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],What is the running time of this algorithm? O(W) O(W) Repeat n times O(n*W) Remember that the brute-force algorithm takes O(2 n )
Example Let’s run our algorithm on the following data: n = 4 (# of elements) W = 5 (max weight) Elements (weight, benefit): (2,3), (3,4), (4,5), (5,6)
Example (2) for w = 0 to W B[0,w] = 0 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 4
Example (3) for i = 0 to n B[i,0] = 0 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 4
Example (4) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 1 w-w i =-1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0
Example (5) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 2 w-w i =0 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3
Example (6) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 3 w-w i =1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3
Example (7) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 4 w-w i =2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3
Example (8) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 5 w-w i =2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3
Example (9) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 1 w-w i =-2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0
Example (10) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 2 w-w i =-1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3
Example (11) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 3 w-w i =0 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3 4
Example (12) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 4 w-w i =1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3 4 4
Example (13) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 5 w-w i =2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3 4 4 7
Example (14) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 1..3 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 0 3 4 4 7 0 3 4
Example (15) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 4 w- w i =0 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 3 3 3 3
Example (15) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 5 w- w i =1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 7 3 3 3 3
Example (16) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 1..4 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 7 0 3 4 5 3 3 3 3
Example (17) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 5 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 7 0 3 4 5 7 3 3 3 3
Comments ,[object Object],[object Object],[object Object]
Conclusion ,[object Object],[object Object],[object Object],[object Object],[object Object]
The End

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lecture 25

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  • 8. 0-1 Knapsack problem: a picture w i b i 10 9 8 5 5 4 4 3 3 2 Weight Benefit value This is a knapsack Max weight: W = 20 Items W = 20
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  • 13. Defining a Subproblem Max weight: W = 20 For S 4 : Total weight: 14; total benefit: 20 w i b i 10 8 5 5 4 4 3 3 2 Weight Benefit 9 Item # 4 3 2 1 5 S 4 S 5 For S 5 : Total weight: 20 total benefit: 26 Solution for S 4 is not part of the solution for S 5 !!! ? w 1 =2 b 1 =3 w 2 =4 b 2 =5 w 3 =5 b 3 =8 w 4 =3 b 4 =4 w 1 =2 b 1 =3 w 2 =4 b 2 =5 w 3 =5 b 3 =8 w 4 =9 b 4 =10
  • 14.
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  • 19. Example Let’s run our algorithm on the following data: n = 4 (# of elements) W = 5 (max weight) Elements (weight, benefit): (2,3), (3,4), (4,5), (5,6)
  • 20. Example (2) for w = 0 to W B[0,w] = 0 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 4
  • 21. Example (3) for i = 0 to n B[i,0] = 0 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 4
  • 22. Example (4) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 1 w-w i =-1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0
  • 23. Example (5) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 2 w-w i =0 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3
  • 24. Example (6) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 3 w-w i =1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3
  • 25. Example (7) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 4 w-w i =2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3
  • 26. Example (8) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=1 b i =3 w i =2 w= 5 w-w i =2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3
  • 27. Example (9) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 1 w-w i =-2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0
  • 28. Example (10) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 2 w-w i =-1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3
  • 29. Example (11) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 3 w-w i =0 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3 4
  • 30. Example (12) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 4 w-w i =1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3 4 4
  • 31. Example (13) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=2 b i =4 w i =3 w= 5 w-w i =2 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 3 4 4 7
  • 32. Example (14) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 1..3 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 3 3 3 3 0 0 3 4 4 7 0 3 4
  • 33. Example (15) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 4 w- w i =0 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 3 3 3 3
  • 34. Example (15) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 5 w- w i =1 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 7 3 3 3 3
  • 35. Example (16) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 1..4 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 7 0 3 4 5 3 3 3 3
  • 36. Example (17) if w i <= w // item i can be part of the solution if b i + B[i-1,w-w i ] > B[i-1,w] B[i,w] = b i + B[i-1,w- w i ] else B[i,w] = B[i-1,w] else B[i,w] = B[i-1,w] // w i > w 0 0 0 0 0 0 W 0 1 2 3 4 5 i 0 1 2 3 0 0 0 0 i=3 b i =5 w i =4 w= 5 Items: 1: (2,3) 2: (3,4) 3: (4,5) 4: (5,6) 4 0 0 0 3 4 4 7 0 3 4 5 7 0 3 4 5 7 3 3 3 3
  • 37.
  • 38.