Star-Based Separators for Intersection Graphs of $c$-Colored Pseudo-Segments

TL;DR

The paper introduces star-based (biclique-based) separators for geometric intersection graphs, addressing the limitation that segment graphs lack small clique-based separators due to large bicliques. It proves that a star-based separator of size $O(\sqrt{n})$ exists for $c$-colored pseudo-segments (and $c$-oriented polygons) and provides an $O(n\log n)$-time construction. This leads to an almost-exact distance oracle for these intersection graphs with $O(n\sqrt{n})$ storage and $O(\sqrt{n})$ query time, achieving subquadratic storage and sublinear query time with an additive error of at most 2. The results extend the separator toolkit beyond disks to non-disk-like objects, enabling faster algorithms and practical distance queries in a broad class of geometric intersection graphs.

Abstract

The Planar Separator Theorem, which states that any planar graph $\mathcal{G}$ has a separator consisting of $O(\sqrt{n})$ nodes whose removal partitions $\mathcal{G}$ into components of size at most $\tfrac{2n}{3}$, is a widely used tool to obtain fast algorithms on planar graphs. Intersection graphs of disks, which generalize planar graphs, do not admit such separators. It has recently been shown that disk graphs do admit so-called clique-based separators that consist of $O(\sqrt{n})$ cliques. This result has been generalized to intersection graphs of various other types of disk-like objects. Unfortunately, segment intersection graphs do not admit small clique-based separators, because they can contain arbitrarily large bicliques. This is true even in the simple case of axis-aligned segments. In this paper we therefore introduce biclique-based separators (and, in particular, star-based separators), which are separators consisting of a small number of bicliques (or stars). We prove that any $c$-oriented set of $n$ segments in the plane, where $c$ is a constant, admits a star-based separator consisting of $O(\sqrt{n})$ stars. In fact, our result is more general, as it applies to any set of $n$ pseudo-segments that is partitioned into $c$ subsets such that the pseudo-segments in the same subset are pairwise disjoint. We extend our result to intersection graphs of $c$-oriented polygons. These results immediately lead to an almost-exact distance oracle for such intersection graphs, which has $O(n\sqrt{n})$ storage and $O(\sqrt{n})$ query time, and that can report the hop-distance between any two query nodes in the intersection graph with an additive error of at most 2. This is the first distance oracle for such types of intersection graphs that has subquadratic storage and sublinear query time and that only has an additive error.

Figures (1)

• Figure 1: Left: A $3$-colored set of pseudo-segments. Middle: The active fragments created by our algorithm. Right: The contact graph induced by the active fragments.