SIMPLE ALGORITHMS FOR ENUMERATING INTERPOINT DISTANCES AND FINDING k NEAREST NEIGHBORS

We present an O(n log n k log k) time and O(n k) space algorithm which takes as input a set of n points in the plane and enumerates the k smallest distances between pairs of points in nondecreasing order. We also present an O(n log n + kn log k) solution to the problem of finding the k nearest neighbors for each of n points. Both algorithms are conceptually very simple, are easy to implement, and are based on a common data structure: the Delaunay triangulation. Variants of the algorithms work for any convex distance function metric.