Balanced Line Separators of Unit Disk Graphs

TL;DR

This paper establishes a geometric graph separator theorem for unit disk intersection graphs, providing lines that partition disks with bounded intersections and disk counts, along with algorithms and experimental validation.

Contribution

It introduces a new separator theorem for unit disk graphs with constructive, linear-time algorithms and demonstrates improved practical performance over previous methods.

Findings

Existence of a line intersecting O(√(m+n) log n) disks with balanced halves.

An axis-parallel line intersecting O(√(m+n)) disks with less balanced halves.

Experimental results show the method outperforms previous separator algorithms.

Abstract

We prove a geometric version of the graph separator theorem for the unit disk intersection graph: for any set of $n$ unit disks in the plane there exists a line $\ell$ such that $\ell$ intersects at most $O(\sqrt{(m+n)\log{n}})$ disks and each of the halfplanes determined by $\ell$ contains at most $2n/3$ unit disks from the set, where $m$ is the number of intersecting pairs of disks. We also show that an axis-parallel line intersecting $O(\sqrt{m+n})$ disks exists, but each halfplane may contain up to $4n/5$ disks. We give an almost tight lower bound (up to sublogarithmic factors) for our approach, and also show that no line-separator of sublinear size in $n$ exists when we look at disks of arbitrary radii, even when $m=0$. Proofs are constructive and suggest simple algorithms that run in linear time. Experimental evaluation has also been conducted, which shows that for random instances our method outperforms the method by Fox and Pach (whose separator has size $O(\sqrt{m})$).