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

Abstract

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalisation of the concept of the median of a (finite) set of points on the real line. In this paper, we present an algorithm for computing a centerpoint of a set of n points in the plane. The algorithm has complexity O(n) which significantly improves the O(n log3 n) complexity of the previously best known algorithm. We use suitable modifications of the ham-sandwich-cut algorithm and the prune-and-search technique to achieve this improvement.