How to find two most distant points?
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.
Boundary point algorithms abound (look for convex hull algorithms). From there, it should take O(N) time to find the most-distant opposite points.
From the author's comment: first find any pair of opposite points on the hull, and then walk around it in semi-lock-step fashion. Depending on the angles between edges, you will have to advance either one walker or the other, but it will always take O(N) to circumnavigate the hull.
For this specific problem, with just a list of Euclidean points, one way is to find the convex hull of the set of points. The two distant points can then be found by traversing the hull once with the rotating calipers method.
Here is an O(N log N) implementation:
- http://mukeshiiitm.wordpress.com/2008/05/27/find-the-farthest-pair-of-points/
If the list of points is already sorted, you can remove the sort to get the optimal O(N) complexity.
For a more general problem of finding most distant points in a graph: Algorithm to find two points furthest away from each other
The accepted answer works in O(N^2).