You're are working on n different projects, but in m hours you are going on vacation. Imagine that for each project i, you had a function fi that told you how happy the people paying for project i would be if out your m available hours you devoted 0 = x = m to project i.
Imagine further that each such function has maximum value at most s, corresponding to the project being fully finished (and thus the clients being perfectly happy).
Give a dynamic programming for determining the most happiness you can generate for your clients by splitting your m hours among the n projects.
• Because of book-keeping rules, you can't spend fractional hours on a project.
• The functions fi are non-decreasing, i.e., for every project i, if t1 = t2, then fi(t1) = fi(t2).
• The running time of your algorithm must be O(nambsc) for some fixed integers a, b, c = 0.
For this problem:
1. Write an identity for the OPT value.
2. Explain why the idenity holds. Your explanation should appear self-evident.
3. Describe the data structure you will use to store the OPT value for the subproblems and the order in which you will fill out the entries in your data structure.
4. In the problems you are asked to return the solution (not just the value of the solution), state the additional information you record which will allow you to retrace your steps and report an optimal solution.
5. Bound the running time by the size of the data structure and the work per entry in it.