AnO(n logn) algorithm for the all-nearest-neighbors Problem

Given a setV ofn points ink-dimensional space, and anL _q -metric (Minkowski metric), the all-nearest-neighbors problem is defined as follows: for each pointp inV, find all those points inV–{p} that are closest top under the distance metricL _q . We give anO(n logn) algorithm for the all-nearest-neighbors problem, for fixed dimensionk and fixed metricL _q . Since there is an (n logn) lower bound, in the algebraic decision-tree model of computation, on the time complexity of any algorithm that solves the all-nearest-neighbors problem (fork=1), the running time of our algorithm is optimal up to a constant factor.This research was supported by a fellowship from the Shell Foundation. The author is currently at AT&T Bell Laboratories, Murray Hill, New Jersey, USA.