The largest empty annulus problem

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, that partitions S into a subset of points inside and a subset of points outside the circles, such that no point p ∈ S lies in the interior of A. This problem can be considered as a maximin 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 that 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, furthermore, we can simultaneously optimize for all values of k within the same time bound. When k is small, that is a fixed constant, we can solve the problem in O(n log n) time and O(n) space ​
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