Maximization using Dynamic Programming

Question

I am trying to come up with the solution for a problem analogous to the following:

• Let M be a matrix of n rows and T columns.
• Let each row have positive non-decreasing values. (e.g. row = [1, 2, 30, 30, 35])
• Let M[i][j] correspond to the score obtained by spending j units of time on exam i.

Using dynamic programming, solve the problem as to find the optimal way of spending T units of time to study which will yield the highest total score.

Thanks in advance for any help :)

My attempt:

S[][] = 0

for i = 1:n
   for j = 0:T
       max = 0
       for k = 0:j
           Grade = G[i][j]+ S[i-1][T-k]
           if Grade > max
              max = Grade
       end for
       S[i][j] = max
    end for
end for

Answer 1

Let S[i][j] represent the best score you can achieve spending j units of time on the first i exams. You can calculate S[i][j] by looking at S[i-1][k] for each value of k. For each element of S, remember the value of k from the previous row that gave the best result. The answer to what the best score for studying all exams in time T is just S[n][T], and you can use the values of k that you remembered to determine how much time to spend on each exam.

S[][] = 0

for j = 0:T
   S[0][j] = M[0][j]

for i = 1:n
   for j = 0:T
       max = 0
       for k = 0:j
           # This is the score you could get by spending j time, spending
           # k time on this exam and j-k time on the previous exams.
           Grade = S[i-1][j-k] + M[i][k]
           if Grade > max
              max = Grade
       end for
       S[i][j] = max
    end for
end for