How to find two most distant points?

Question

This is a question that I was asked on a job interview some time ago. And I still can\'t figure out sensible answer.

Question is:

you are given set of points

Answer 1

You are looking for an algorithm to compute the diameter of a set of points, Diam(S). It can be shown that this is the same as the diameter of the convex hull of S, Diam(S) = Diam(CH(S)). So first compute the convex hull of the set.

Now you have to find all the antipodal points on the convex hull and pick the pair with maximum distance. There are O(n) antipodal points on a convex polygon. So this gives a O(n lg n) algorithm for finding the farthest points.

This technique is known as Rotating Calipers. This is what Marcelo Cantos describes in his answer.

If you write the algorithm carefully, you can do without computing angles. For details, check this URL.