License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.54
URN: urn:nbn:de:0030-drops-59467
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5946/
Kupavskii, Andrey ;
Mustafa, Nabil ;
Pach, János
New Lower Bounds for epsilon-Nets
Abstract
Following groundbreaking work by Haussler and Welzl (1987), the use of small epsilon-nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest epsilon-nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Konemann, and Sharpe); and (ii) the construction of a set system whose members can be obtained by intersecting a point set in R^4 by a family of half-spaces such that the size of any epsilon-net for them is at least (1/(9*epsilon)) log (1/epsilon) (Pach and Tardos).
The present paper completes both of these avenues of research. We (i) give a lower bound, matching the result of Chan et al., and (ii) generalize the construction of Pach and Tardos to half-spaces in R^d, for any d >= 4, to show that the general upper bound of Haussler and Welzl for the size of the smallest epsilon-nets is tight.
BibTeX - Entry
@InProceedings{kupavskii_et_al:LIPIcs:2016:5946,
author = {Andrey Kupavskii and Nabil Mustafa and J{\'a}nos Pach},
title = {{New Lower Bounds for epsilon-Nets}},
booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)},
pages = {54:1--54:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-009-5},
ISSN = {1868-8969},
year = {2016},
volume = {51},
editor = {S{\'a}ndor Fekete and Anna Lubiw},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/5946},
URN = {urn:nbn:de:0030-drops-59467},
doi = {10.4230/LIPIcs.SoCG.2016.54},
annote = {Keywords: epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces}
}
Keywords: |
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epsilon-nets; lower bounds; geometric set systems; shallow-cell complexity; half-spaces |
Collection: |
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32nd International Symposium on Computational Geometry (SoCG 2016) |
Issue Date: |
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2016 |
Date of publication: |
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10.06.2016 |