Dynamic programming question

Question

A circus is designing a tower routine consisting of people standing atop one another’s shoulders. For practical and aesthetic reasons, each person must be both shorter and light

Answer 1

You're absolutely correct. Doing just one direction is enough.

A proof is easy by using the maximum property of the subsequence. We assume one side (say the left) of values is ordered, and take the longest descending subsequence of the right. We now perform the other operation, order the right and take the subsequence from the left.

If we arrive at a list that is either shorter or longer than the first one we found we have reached a contradiction, since that subsequence was ordered in the very same relative order in the first operation, and thus we could have found a longer descending subsequence, in contradiction to the assumption that the one we took was maximal. Similarly if it's shorter then the argument is symmetrical.

We conclude that finding the maximum on just one side will be the same as the maximum of the reverse ordered operation.

Worth noting that I haven't proven that this is a solution to the problem, just that the one-sided algorithm is equivalent to the two-sided version. Although the proof that this is correct is almost identical, assume that there exists a longer solution and it contradicts the maximalness of the subsequence. That proves that there is nothing longer, and it's trivial to see that every solution the algorithm produces is a valid solution. Which means the algorithm's result is both >= the solution and <= the solution, therefore it is the solution.