eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-06-09
23:1
23:16
10.4230/LIPIcs.SoCG.2023.23
article
Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size
Chan, Timothy M.
1
https://orcid.org/0000-0002-8093-0675
Huang, Zhengcheng
1
Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
In SoCG 2022, Conroy and Tóth presented several constructions of sparse, low-hop spanners in geometric intersection graphs, including an O(nlog n)-size 3-hop spanner for n disks (or fat convex objects) in the plane, and an O(nlog² n)-size 3-hop spanner for n axis-aligned rectangles in the plane. Their work left open two major questions: (i) can the size be made closer to linear by allowing larger constant stretch? and (ii) can near-linear size be achieved for more general classes of intersection graphs?
We address both questions simultaneously, by presenting new constructions of constant-hop spanners that have almost linear size and that hold for a much larger class of intersection graphs. More precisely, we prove the existence of an O(1)-hop spanner for arbitrary string graphs with O(nα_k(n)) size for any constant k, where α_k(n) denotes the k-th function in the inverse Ackermann hierarchy. We similarly prove the existence of an O(1)-hop spanner for intersection graphs of d-dimensional fat objects with O(nα_k(n)) size for any constant k and d.
We also improve on some of Conroy and Tóth’s specific previous results, in either the number of hops or the size: we describe an O(nlog n)-size 2-hop spanner for disks (or more generally objects with linear union complexity) in the plane, and an O(nlog n)-size 3-hop spanner for axis-aligned rectangles in the plane.
Our proofs are all simple, using separator theorems, recursion, shifted quadtrees, and shallow cuttings.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol258-socg2023/LIPIcs.SoCG.2023.23/LIPIcs.SoCG.2023.23.pdf
Hop spanners
geometric intersection graphs
string graphs
fat objects
separators
shallow cuttings