The first line contains a single integer T representing the number of test cases. The T-test cases are as follows: Line 1:The first line contains an integer, that denotes the value of N. Line 2:The following line contains N space-separated integers, that denote the values of the weight of items. Line 3:The following line contains N space-separated integers, that denote the values associated with the items. Line 4:The following line contains an integer that denotes the value of W. W denotes the maximum weight that a thief can carry.
The first and only line of output contains the maximum value that a thief can generate, as described in the task. The output of every test case is printed in a separate line.
1 <= T <= 10 1 <= N <= 10^2 1<= wi <= 50 1 <= vi <= 10^2 1 <= W <= 10^3 Time Limit: 1 second
Can we solve this using space complexity of not more than O(W)?
Approach: In the Dynamic programming we will work considering the same cases as mentioned in the recursive approach. In a DP table let’s consider all the possible weights from ‘1’ to ‘W’ as the columns and weights that can be kept as the rows.
The state DP[i][j] will denote maximum value of ‘j-weight’ considering all values from ‘1 to ith’. So if we consider ‘wi’ (weight in ‘ith’ row) we can fill it in all columns which have ‘weight values > wi’. Now two possibilities can take place:
Now we have to take a maximum of these two possibilities, formally if we do not fill ‘ith’ weight in ‘jth’ column then DP[i][j] state will be same as DP[i-1][j] but if we fill the weight, DP[i][j] will be equal to the value of ‘wi’+ value of the column weighing ‘j-wi’ in the previous row. So we take the maximum of these two possibilities to fill the current state. This visualization will make the concept clear: