Approximate center points in dense point sets

Abstract

The notion of center points is one way to generalize the median of a finite ordered point set to higher dimensions. A set P of n points in d-dimensional Euclidean space E d is δ- dense, if the ratio of the largest to the smallest distance between any two points of P is bounded by δn 1 d , with some constant δ. We describe a simple, yet efficient algorithm to compute an approximate center point for a δ-dense set P in time O( dn). The quality of the approximation depends on δ and exponentially on the dimension d, where the absolute value of the exponent is in Ω(d log d). We also present an iteration method with a linear rate of convergence, which computes an improved 1 (2(d − 1)(δ + 1) d) - center point for sufficiently large n.