Separating Balls with a Hyperplane

Let D be a set of n pairwise disjoint unit balls in R 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, which correspond to n/2-separators. The analogous notion for point sets has been well studied. Separators have various applications, for instance, in divide-andconquer 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 two deterministic algorithms to bisect or approximately bisect a given set of disjoint unit balls by a hyperplane: Firstly, we present a simple linear-time algorithm to construct an αn-separator for balls in R, for any 0 < α < 1/2, that intersects at most cn(d−1)/d balls, for some constant c that depends on d and α. The number of intersected balls is best possible up to the constant c. Secondly, we present a near-linear time algorithm to construct an (n/2 − o(n))-separator in R that intersects o(n) balls.