Computing the center region and its variants

Abstract

Given a set S of n points in d-dimensional Euclidean space, a point x in the space is called a centerpoint of S if every closed halfspace containing x contains at least ⌈n/(d+1)⌉ points from S. The center region of S is the set of all centerpoints of S in the space. We present an O(n2log4⁡n)-time algorithm for computing the center region for d=3. This improves the previously best known algorithm by Agarwal, Sharir and Welzl, which takes O(n2+ε) time for any ε>0. It is known that the combinatorial complexity of the center region is Ω(n2) in the worst case, thus our algorithm is almost tight. We also consider the problem of computing a colored version of the center region of n points in the two-dimensional Euclidean space and present an O(nlog4⁡n)-time algorithm. For the three-dimensional Euclidean space, we show that the colorful center region of n colored points can be computed in O(n2log4⁡n) time.