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DynProgKnapsack

# DynProgKnapsack - The 0-1 Knapsack Problem The difference...

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The 0-1 Knapsack Problem The difference between this problem and the fractional one is that you can't take a fraction of an item. You either take the whole thing or none of it. So here, is the problem formally described: Your goal is to maximize the value of a knapsack that can hold at most W units worth of goods from a list of items I 0 , I 1 , ... I n- 1 . Each item has two attributes: 1) Value - let this be v i for item I i . 2) Weight - let this be w i for item I i . Now, instead of being able to take a certain weight of an item, you can only either take the item or not take the item. The naive way to solve this problem is to cycle through all 2 n subsets of the n items and pick the subset with a legal weight that maximizes the value of the knapsack. But, we can find a dynamic programming algorithm that will USUALLY do better than this brute force technique. Our first attempt might be to characterize a sub-problem as follows: Let S k be the optimal subset of elements from {I 0 , I 1 ,... I k }. But what we find is that the optimal subset from the elements {I 0 , I 1 ,... I k+1 } may not correspond to the optimal subset of elements from {I 0 , I 1 ,... I k } in any regular pattern. Basically, the solution

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DynProgKnapsack - The 0-1 Knapsack Problem The difference...

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