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.