2026-07-24T01:13:37Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:13827 2024-03-06T10:52:49Z ddc:004 open_access

Escaping the Curse of Spatial Partitioning: Matchings with Low Crossing Numbers and Their Applications Csikós, Mónika Mustafa, Nabil H. Matchings crossing numbers approximations 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 𝒮 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Mónika Csikós and Nabil H. Mustafa 2021 Is Part Of LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2021.28 urn:nbn:de:0030-drops-138273 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.28 eng https://creativecommons.org/licenses/by/4.0/legalcode