Halving Balls in Deterministic Linear Time

Abstract

Let \({\mathcal{D}}\) be a set of n pairwise disjoint unit balls in ℝ d and P the set of their center points. A hyperplane \({\mathcal{H}}\) is an m-separator for \({\mathcal{D}}\) if each closed halfspace bounded by \({\mathcal{H}}\) 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 < α < 1/2, that intersects at most cn (d − 1)/d balls, where c depends on d and α. The number of balls intersected is best possible up to the constant c. Secondly, we present a near-linear time algorithm to find an (n/2 − o(n))-separator in ℝ d that intersects o(n) balls. Finally, we give a linear-time algorithm to construct a halving line in ℝ2 that intersects O(n (5/6) + ε ) disks.Our results improve the runtime of a disk sliding algorithm by Bereg, Dumitrescu and Pach. In addition, our results improve and derandomize an algorithm to construct a space decomposition used by Löffler and Mulzer to construct an onion decomposition for imprecise points.