Balanced Line Separators of Unit Disk Graphs

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 $O(\sqrt{m})$).

Chiu, van Renssen and Roeloffzen were supported by JST ERATO Grant Number JPMJER1305, Japan. Korman was supported in part by KAKENHI Nos. 12H00855 and 17K12635. Katz was partially supported by grant 1884/16 from the Israel Science Foundation. Okamoto was partially supported by KAKENHI Grant Numbers JP24106005, JP24220003 and JP15K00009, JST CREST Grant Number JPMJCR1402, and Kayamori Foundation for Informational Science Advancement. Smorodinsky’s research was partially supported by Grant 635/16 from the Israel Science Foundation.