Algorithms for Proximity Problems in Higher Dimensions

We present algorithms for five interdistance enumeration problems that take as input a set S of n points in IRd (for a fixed but arbitrary dimension d) and as output enumerate pairs of points in S satisfying various conditions. We present: an O(n log n + k) time and O(n) space algorithm that takes as additional input a distance δ and outputs all k pairs of points in S separated by a distance of δ or less; an O(n log n + k log k) time and O(n+k) space algorithm that enumerates in non-decreasing order the k closest pairs of points in S; an O(n log n + k) time algorithm for the same problem without any order restrictions; an O(nk log n) time and O(n) space algorithm that enumerates in nondecreasing order the nk pairs representing the k nearest neighbors of each point in S; and an O(n log n+ kn) time algorithm for the same problem without any order restrictions. The algorithms combine a modification of the planar approach of Dickerson, Drysdale, and Sack [11] with the method of Bern, Eppstein, and Gilbert [3] for augmenting a point set to have a linear size bounded degree Delaunay triangulation. Thus, in addition to providing new solutions to these problems, the paper also shows how the Delaunay triangulation can be used as the underlying data structure in a unified approach to proximity problems even in higher dimensions.