Halving Balls in Deterministic Linear Time

TL;DR

This paper introduces three deterministic algorithms for efficiently bisecting sets of disjoint unit balls in various dimensions, optimizing the number of intersected balls and improving previous computational methods.

Contribution

The paper presents new linear-time and near-linear algorithms for constructing hyperplane separators with minimal intersections, advancing deterministic solutions in high-dimensional geometric partitioning.

Findings

Constructed an $eta n$-separator intersecting at most $cn^{(d-1)/d}$ balls in linear time.

Developed a near-linear time algorithm for an $(n/2 - o(n))$-separator intersecting $o(n)$ balls.

Designed a linear-time halving line in $ ^2$ intersecting $O(n^{(5/6)+ ext{ extepsilon}})$ disks.

Abstract

Let $\D$ be a set of $n$ pairwise disjoint unit balls in $\R^d$ 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 $\alpha n$-separator for balls in $\R^d$, for any $0<\alpha<1/2$, that intersects at most $cn^{(d-1)/d}$ balls, for some constant $c$ that depends on $d$ and $\alpha$. 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 $\R^d$ that intersects $o(n)$ balls. Finally, we give a linear-time algorithm to construct a halving line in $\R^2$ that intersects $O(n^{(5/6)+\epsilon})$ 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{\"o}ffler and Mulzer to construct an onion (convex layer) decomposition for imprecise points (any point resides at an unknown location within a given disk).