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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.2021.28
URN: urn:nbn:de:0030-drops-138273
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13827/
Csikós, Mónika ;
Mustafa, Nabil H.
Escaping the Curse of Spatial Partitioning: Matchings with Low Crossing Numbers and Their Applications
Abstract
Given a set system (X, S), constructing a matching of X with low crossing number is a key tool in combinatorics and algorithms. In this paper we present a new sampling-based algorithm which is applicable to finite set systems. Let n = |X|, m = | S| and assume that X has a perfect matching M such that any set in ? crosses at most κ = Θ(n^γ) edges of M. In the case γ = 1- 1/d, our algorithm computes a perfect matching of X with expected crossing number at most 10 κ, in expected time Õ (n^{2+(2/d)} + mn^(2/d)).
As an immediate consequence, we get improved bounds for constructing low-crossing matchings for a slew of both abstract and geometric problems, including many basic geometric set systems (e.g., balls in ℝ^d). This further implies improved algorithms for many well-studied problems such as construction of ε-approximations. Our work is related to two earlier themes: the work of Varadarajan (STOC '10) / Chan et al. (SODA '12) that avoids spatial partitionings for constructing ε-nets, and of Chan (DCG '12) that gives an optimal algorithm for matchings with respect to hyperplanes in ℝ^d.
Another major advantage of our method is its simplicity. An implementation of a variant of our algorithm in C++ is available on Github; it is approximately 200 lines of basic code without any non-trivial data-structure. Since the start of the study of matchings with low-crossing numbers with respect to half-spaces in the 1980s, this is the first implementation made possible for dimensions larger than 2.
BibTeX - Entry
@InProceedings{csikos_et_al:LIPIcs.SoCG.2021.28,
author = {Csik\'{o}s, M\'{o}nika and Mustafa, Nabil H.},
title = {{Escaping the Curse of Spatial Partitioning: Matchings with Low Crossing Numbers and Their Applications}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {28:1--28:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-184-9},
ISSN = {1868-8969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13827},
URN = {urn:nbn:de:0030-drops-138273},
doi = {10.4230/LIPIcs.SoCG.2021.28},
annote = {Keywords: Matchings, crossing numbers, approximations}
}
