Algorithm for Human Towering

Question

In Cracking the Coding Interview, Fourth Edition, there is such a problem:

A circus is designing a tower routine consisting of people standing atop

Answer 1

You can solve the problem with dynamic programming.

Sort the troupe by height. For simplicity, assume all the heights h_i and weights w_j are distinct. Thus h_i is an increasing sequence.

We compute a sequence T_i, where T_i is a tower with person i at the top of maximal size. T_1 is simply {1}. We can deduce subsequent T_k from the earlier T_j — find the largest tower T_j that can take k's weight (w_j < w_k) and stand k on it.

The largest possible tower from the troupe is then the largest of the T_i.

This algorithm takes O(n**2) time, where n is the cardinality of the troupe.