Halving Balls in Deterministic Linear Time

Abstract

Let $\D$ be a set of $n$ pairwise disjoint unit balls in ${\mathbb{R}}^d$ and $P$ the set of their center points. A hyperplane $\Hy$ is an \emph{$m$-separator} for $\D$ if each closed halfspace bounded by $\Hy$ 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-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 disjoint unit balls by a hyperplane: Firstly, we present a simple linear-time algorithm to construct an $\hat{I}\pm n$-separator for balls in ${\mathbb{R}}^d$, for any $0<\hat{I}\pm <1/2$, that intersects at most $c{n}^{(d-1)/d}$ balls, for some constant $c$ that depends on $d$ and $\hat{I}\pm$. 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 ${\mathbb{R}}^d$ that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to construct a halving line in ${\mathbb{R}}^2$ that intersects $O(n^{(5/6)+\hat{I}\mu })$ 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 L{ö}ffler and Mulzer to construct an onion (convex layer) decomposition for imprecise points (any point resides at an unknown location within a given disk).