1. Knapsack Problem Section 7.6

2. Problem Suppose we have n items U={u 1,..u n }, that we would like to insert into a knapsack of size C. Each item u i has a size s i and a value v i. A choose a subset of items S  U such that

3. Dynamic Programming Solution For 1  j  C & 1  I  n, let V[i,j] be the optimal value obtained by filling a knapsack of size j with items taken from {u 1,..u i }. Also V[i,0]=0 and V[0,j]=0. Our Goal is to find V[n,C]

4. Recursive Formula

5. Algorithm: Knapsack Input: Knapsack of size C, items U={u 1,..u n }, with sizes s 1,.., s n, and values v 1,.., v n Output: For i  0 to n V[i,0]  0 end for For j  0 to C V[0,j]  0 end for for i  1 to n for j  1 to C V[i,j] = V[i-1,j] if s i  j then V[i,j] =max{V[i,j], V[i-1,j- s i ] + v i } end for Return V[n,C]

6. Performance Analysis Time =  (nC) Space =  (nC) But can be modified to use only  (C) space by storing two rows only.

7. How does it work? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000000000000 0 1 2 3.......................s1...................s2.............................................C 012.....n012.....n 0 0 0 0 0 0 0 0 0 0 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 Copy the above red line here = max{ + v2, }

8. How does it work? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000000000000 0 1 2 3.......................s1...................s2.............................................C 012.....n012.....n 0 0 0 0 0 0 0 0 0 0 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 Copy the above red line here = max{ + v2, }

9. How does it work? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000000000000 0 1 2 3.......................s1...................s2.............................................C 012.....n012.....n 0 0 0 0 0 0 0 0 0 0 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 Copy the above red line here = max{ + v2, }

10. How does it work? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000000000000 0 1 2 3.......................s1...................s2.............................................C 012.....n012.....n 0 0 0 0 0 0 0 0 0 0 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 Copy the above red line here = max{ + v2, }

11. How does it work? 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0000000000000000 0 1 2 3.......................s1...................s2.............................................C 012.....n012.....n 0 0 0 0 0 0 0 0 0 0 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 v1 Copy the above red line here = max{ + v2, }