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.

Partially supported by the ESF EUROCORES programme EuroGIGA, CRP GraDR and the Swiss National Science Foundation, SNF Project 20GG21-134306.