Help writing a dynamic programming algorithm in English

Question

It’s Friday night and you have $n$ parties to go to. Party $i$ has a start time $s_i$ end time $t_i$ and a value $v_i \ge 0$. Think of the value as an indicator of how excited you are about that party. You want to pick a subset of the parties to attend so that you get maximum total value. The only constraint is that in order to get a value $v_i$ from party $i$ you need to attend the party from start to finish. Hence you cannot drop in midway and leave before the party ends. This means that if two parties overlap, you can only attend one of them. For example, if party 1 has a start time of 7pm and ends at 9pm and party 2 starts at 7:30pm and ends at 10pm, you can only attend one of them. On the other hand, if party 2 starts at 9pm or later then you can attend both. Given as input start times, end times and values of each of the n parties, design an $O(n \log n)$ time algorithm to plan your night optimally. Describe in English the idea of your algorithm. If you use dynamic programming define the memo table, base case and recursive steps. You don’t need to write the pseudo code or provide a proof of correctness.

I am having difficulty beginning this problem. My initial thought was to have the base case be to choose the value with the initial start time and the greatest $v_i$, however if party $i$ runs for your entire start to end time, you might not maximize $v_i$. I would greatly appreciate some insight into how to do this problem.

Answer 1

Since this problem is a learning exercise of yours, I will just write the ordered subproblems used in dynamic programming. In general, I consider it is about more than half way through a nontrivial problem that can be solved by dynamic programming when the ordered subproblems have been specified clearly, at least in term of "difficult" thinking since the rest are more routine.

The subproblems are computing $v[i]$, the maximal total value you can get when you just finished party $P_i$. To help computing these subproblems, that is, from earlier subproblems to later subproblems, you may sort the parties by their ending times so that $P_1,P_2,\cdots,P_n$ end in order of time. After the sorting, $v[i]$ will only depend on those $v[j]$ where $j<i$ or, to be more precise, those $P_j$ whose ending time is before $P_i$'s starting time. (This condition can be made even more restrictive, which may be unnecessary for your exercise and which may be necessary to get a better grade.) There are several more details such as the base case and the recurrence relation, which I will leave for you to fill in.