Computing the Center Region and Its Variants

10/27/2019

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by   Eunjin Oh, et al.

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We present an O(n^2log^4 n)-time algorithm for computing the center region of a set of n points in the three-dimensional Euclidean space. This improves the previously best known algorithm by Agarwal, Sharir and Welzl, which takes O(n^2+ϵ) time for any ϵ > 0. It is known that the combinatorial complexity of the center region is Ω(n^2) in the worst case, thus our algorithm is almost tight. We also consider the problem of computing a colored version of the center region in the two-dimensional Euclidean space and present an O(nlog^4 n)-time algorithm.