1. Knapsack Problem An Example of Dynamic Programming

2. Problem Definition A small knapsack of capacity M N types of items of varying size and value Find the combination of items to maximize the total value

3. Example NameABCDE Size34789 Value45101113 What is the best combination for size 17?

4. Strategy: calculate all sizes up to M for (j = 1; j <= N; j++) { for (i = 1; i <= M; i++) { if (i >= size[j] && cost[i] < cost[i-size[j]]+val[j]) { cost[i] = cost[i-size[j]]+val[j]; best[i] = j; }

5. k34567891011121314151617 cost[k]44488812 16 20 best[k]AAAAAAAAAAAAAAA cost[k]4558910121314161718202122 best[k]ABBABBABBABBABB cost[k]455810 121415161820 2224 best[k]ABBACBACCACCACC cost[k]45581011121415161820212224 best[k]ABBACDACCACCDCC cost[k]45581011131415171820212324 best[k]ABBACDECCECCDEC