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CS331 Advanced Data Structures – Spring 2016 Knapsack Problem The Problem : Given n value a1 , a2 , . . . , an and a knapsack size K, find a subset of values whose sum is equal to K, i.e. a subset S where X ai = K ai ∈S The Recurrence Relation : Let Ci,k k. Then true, false, true, Ci,k = = true if there is a subset of the first i items that sums to i = 0, k = 0, i = 0, k > 0, i > 0, k ≥ 0 i > 0, k ≥ ai false, otherwise, and Ci−1,k is true, or and Ci−1,k−ai is true, The Code : boolean knapsack(int [] a, int K) { int n = a.length(); boolean [][] table = new boolean[K+1][n+1]; table[0][0] = true; for(int k=1; k<=K; i++) table[k][0] = false; for(int i=1; i<=n; i++) { for(int k=0; k<a[j]; k++) table[k][i] = table[k][i-1]; for(int k=a[j]; k<=K; k++) table[k][i] = table[k][i-1] || table[k-a[j]][i-1]; } return table[K][n]; } The Time Complexity : T (n) = O(nK) (but not polynomial in input size!).