License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.23
URN: urn:nbn:de:0030-drops-178738
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17873/
Chan, Timothy M. ;
Huang, Zhengcheng
Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size
Abstract
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.
BibTeX - Entry
@InProceedings{chan_et_al:LIPIcs.SoCG.2023.23,
author = {Chan, Timothy M. and Huang, Zhengcheng},
title = {{Constant-Hop Spanners for More Geometric Intersection Graphs, with Even Smaller Size}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {23:1--23:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-273-0},
ISSN = {1868-8969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17873},
URN = {urn:nbn:de:0030-drops-178738},
doi = {10.4230/LIPIcs.SoCG.2023.23},
annote = {Keywords: Hop spanners, geometric intersection graphs, string graphs, fat objects, separators, shallow cuttings}
}
|
Keywords: |
|
Hop spanners, geometric intersection graphs, string graphs, fat objects, separators, shallow cuttings |
|
Collection: |
|
39th International Symposium on Computational Geometry (SoCG 2023) |
|
Issue Date: |
|
2023 |
|
Date of publication: |
|
09.06.2023 |
