research-article

Tight lower bounds for the size of epsilon-nets

Authors:

Gábor TardosAuthors Info & Claims

SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry

Pages 458 - 463

https://doi.org/10.1145/1998196.1998271

Published: 13 June 2011 Publication History

Abstract

According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size O(1/ε log 1/ε). Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1/ε, were found by Alon (2010). In his examples, every ε-net is of size Ω(1/εg(1/ε)), where g is an extremely slowly growing function, related to the inverse Ackermann function.

We show that there exist geometrically defined range spaces, already of VC-dimension 2, in which the size of the smallest ε-nets is Ω(1/ε log 1/ε). We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is Ω(1/ε log log 1/ε). By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

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Index Terms

Tight lower bounds for the size of epsilon-nets
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms

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cover image ACM Conferences

SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry

June 2011

532 pages

ISBN:9781450306829

DOI:10.1145/1998196

Program Chairs:
Ferran Hurtado
Universitat Politècnica de Catalunya
,
Marc van Kreveld
Utrecht University

Copyright © 2011 ACM.

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Published: 13 June 2011

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June 13 - 15, 2011

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Banik ARaman RRay S(2023)On the geometric priority set cover problemComputational Geometry10.1016/j.comgeo.2023.101984112(101984)Online publication date: Jun-2023
https://doi.org/10.1016/j.comgeo.2023.101984
Madireddy RMudgal A(2022)Weighted geometric set cover with rectangles of bounded integer side lengthsDiscrete Applied Mathematics10.1016/j.dam.2022.03.004315:C(36-55)Online publication date: 15-Jul-2022
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