Computing a centerpoint of a finite planar set of points in linear time

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalization of the concept of the median of a set of reals. In this paper we present a linear-time algorithm for computing a centerpoint of a set ofn points in the plane, which is optimal compared with theO(n log³ n) complexity of the previously best-known algorithm. We use suitable modifications of the hamsandwich cut algorithm in Me2] and the prune-and-search technique of Megiddo Me1] to achieve this improvement.