Algorithm to find points that are furthest apart — better than O(n^2)?

Question

In my program, I have a set of points. For purposes of rescaling, I am searching for the two nodes that are furthest apart, and then computing a factor by which to multiply

Answer 1

As mentioned in this answer, you are seeking the "diameter" of the set of N points, a well known problem in computational geometry. There are basically two steps:

1. Find the convex hull of the points. Algorithms exist that are O(N ln N), worst case. In practice, QuickHull is usually a fast choice, although potentially O(N^2) worst case. The QHull implementation is convenient to call from the command line. The CGAL library provides a C++ implementation

2. Antipodal pairs on the convex hull are candidates for furthest points. One can search over the antipodal points using an algorithm like Rotating calipers in O(N) time.

The problem can be generalized to an "all-farthest pairs" problem: for each point i, find the most distant point j---we're now seeking N pairs of points. The solution again uses the convex hull, but now the second part can be done with a matrix searching algorithm.