How would you look at developing an algorithm for this hotel problem?

Question

There is a problem I am working on for a programming course and I am having trouble developing an algorithm to suit the problem. Here it is:

You are

Answer 1

It looks like you can solve this problem with dynamic programming. The subproblem is the following:

d(i) : The minimum penalty possible when travelling from the start to hotel i.

The recursive formula is as follows:

d(0) = 0 where 0 is the starting position.

d(i) = min_{j=0, 1, ... , i-1} ( d(j) + (200-(ai-aj))^2)

The minimum penalty for reaching hotel i is found by trying all stopping places for the previous day, adding today's penalty and taking the minimum of those.

In order to find the path, we store in a separate array (path[]) which hotel we had to travel from in order to achieve the minimum penalty for that particular hotel. By traversing the array backwards (from path[n]) we obtain the path.

The runtime is O(n^2).