The largest empty annulus problem

Abstract

Given a set of n points S in the Euclidean plane, we address the problem of computing an annulus A, (open region between two concentric circles) of largest width such that no point p ∈ S lies in the interior of A. This problem can be considered as a minimax facility location problem for n points such that the facility is a circumference. We give a characterization of the centres of annuli which are locally optimal and we show the the problem can be solved in O (n3 log n) time and O (n) space. We also consider the case in which the number of points in the inner circle is a fixed value k. When k ∈ O (n) our algorithm runs in O (n3 log n) time and O (n) space. However if k is small, that is a fixed constant, we can solve the problem in O (n log n) time and O(n) space. © Springer-Verlag 2002.