Find subset with elements that are furthest apart from eachother

Question

I have an interview question that I can\'t seem to figure out. Given an array of size N, find the subset of size k such that the elements in the subset are the furthest apart fr

Answer 1

This can be solved in polynomial time using DP.

The first step is, as you mentioned, sort the list A. Let X[i,j] be the solution for selecting j elements from first i elements A.

Now, X[i+1, j+1] = max( min( X[k,j], A[i+1]-A[k] ) ) over k<=i.

I will leave initialization step and memorization of subset step for you to work on.

In your example (1,2,6,10) it works the following way:

    1    2   6   10
1   -    -   -    -
2   -    1   5    9
3   -    -   1    4
4   -    -   -    1