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W 11 1 1 1 2 6 2 3 18 5 walk back through the table 4

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Unformatted text preview: of { 1, 2, …, i–1 } using weight limit W- wi and add vi: OPT(i- 1,W- wi)+vi " if i = 0 $0 $ OPT (i, W ) = # OPT (i − 1, W ) if wi > W $ $ max { OPT (i − 1, W ), vi + OPT (i − 1, W − wi ) } otherwise % Knapsack Problem: Bottom- Up Dynamic Programming !   Knapsack. Fill an n- by- W array. Input: n, W, w1,…,wN, v1,…,vN for w = 0 to W M[0, w] = 0 for i = 1 to n for w = 0 to W if wi > w : M[i, w] = M[i-1, w] else : M[i, w] = max (M[i-1, w], vi + M[i-1, w-wi ]) return M[n, W] Knapsack Algorithm W+1 0 2 3 4 5 6 7 8 9 10 11 φ 0 0 0 0 0 0 0 0 0 0 0 0 {1} n+1 1 0 1 1 1 1 1 1 1 1 1 1 1 { 1, 2 } 0 1 6 7 7 7 7 7 7 7 7 7 { 1, 2, 3 } 0 1 6 7 7 18 19 24 25 25 25 25 { 1, 2, 3, 4 } 0 1 6 7 7 18 22 24 28 29 29 40 { 1, 2, 3, 4, 5 } 0 1 6 7 7 18 22 28 29 34 34 40 OPT: 40 Item Value Weight How do we find the choice vector x, in other words the objects picked in the optimum solution? W = 11 1 1 1 2 6 2 3 18 5 Walk back through the table!! 4 22 6 5 28 7 Knapsack Algorithm W+1 0 2 3 4 5 6 7 8 9 10 11 φ 0 0 0 0 0 0 0 0 0 0 0 0 {1} n+1 1 0 1 1 1 1 1 1 1 1 1 1 1 { 1, 2 } 0 1 6 7 7 7 7 7 7 7 7 7 { 1, 2, 3 } 0 1 6 7 7 18 19 24 25 25 25 25 { 1, 2, 3, 4 } 0 1 6 7 7 18 22 24 28 29 29 40 { 1, 2, 3, 4, 5 } 0 1 6 7 7 18 22 28 29 34 34 40 OPT: 40 n=5 Don’t take object 5 Item Weight 1 1 1 2 W = 11 Value 6 2 3 18 5 4 22 6 5 28 7 Knapsack Algorithm W+1 0 2 3 4 5 6 7 8 9 10 11...
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