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Title: Constant-hop spanners for more geometric intersection graphs, with even smaller size
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.  more » « less
Award ID(s):
2224271
PAR ID:
10499275
Author(s) / Creator(s):
;
Editor(s):
Chambers, Erin W.; Gudmundsson, Joachim
Publisher / Repository:
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Date Published:
Journal Name:
Proc. 39th International Symposium on Computational Geometry (SoCG 2023)
ISBN:
978-3-95977-273-0
Page Range / eLocation ID:
23:1-23:16
Subject(s) / Keyword(s):
Hop spanners geometric intersection graphs string graphs fat objects separators shallow cuttings Theory of computation → Computational geometry
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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