PROBLEM LINKS:

Practice
Contest

Author:Vitaliy Herasymiv
Tester:Tasnim Imran Sunny
Editorialist:Tuan Anh

DIFFICULTY:

Easy

PREREQUISITES:

Dynamic Programming

PROBLEM:

There are N questions. The i^th question needs T[i] minutes to learn and will give you C[i] × P[i] points in the tests. You have W minutes to learn the questions. Your job is to find which questions to learn to maximise your score.

EXPLANATION:

We need to choose a sub-set of N questions so that the total learning time does not exceed W and the total socore is as large as possible. With the constraint N ≤ 100 we cannot try out all possible sub-sets since the number of sub-sets can be 2 ¹⁰⁰ which is extremely large.

We need to use dynamic programming in this problem. Let F[i][j] be the maximal score you can get if you spend no more than j minutes to learn some questions amongst the first i questions. Now, there are two cases:

• If you choose to learn the i^th question then you get maximally F[i - 1][j - T[i]] + C[i] × P[i] points. Be careful with the case where j < T[i].
• If you do not learn the i^th question then you get maximally F[i-1][j] points.

AUTHOR'S AND TESTER'S SOLUTIONS:

Author's solution can be found here.
Tester's solution can be found here.

In the solution of problem setter and the tester an optimization is used to reduce the memory complexity. With a little modification of the algorithm above we can solve this problem using a one-dimensional array of W elements. The pseudo code is given below:

    FOR i -> 0 to W
        G[i] = 0
    ENDFOR

    (*)FOR i -> 1 to N
        FOR j -> W downto 0
            G[j] = max(G[j + T[i]], G[j] + C[i] × P[i])
    ENDFOR

After the i^thiteration of the for loop (*), the value in G array is exactly as in the i^th row of the F array in the original algorithm. Notice that in the second For loop we need to consider the value of j in the decreasing order so that each time we use G[j] to update G[j + T[i]] we know that the value in G[j] is the optimized value of using j minutes and the first (i - 1) questions only.