Halving balls in deterministic linear time

Abstract

Let be a set of n pairwise disjoint unit balls in R d and P the set of their center points. A hyperplane H is an m-separator for D if each closed halfspace bounded by H contains at least m points from P. This generalizes the notion of halving hyperplanes (n/2-separators). The analogous notion for point sets has been well studied. Separators have various applications, for instance, in divide-and-conquer schemes. In such a scheme any ball that is intersected by the separating hyperplane may still interact with both sides of the partition. Therefore it is desirable that the separating hyperplane intersects a small number of balls only. We present three deterministic algorithms to bisect or approximately bisect a given set of n disjoint unit balls by a hyperplane: firstly, a linear-time algorithm to construct an αn-separator in R d, for 0