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Discrete & Computational Geometry

, Volume 7, Issue 2, pp 163–173 | Cite as

Almost tight bounds forɛ-Nets

  • János Komlós
  • János Pach
  • Gerhard Woeginger
Article

Abstract

Given any natural numberd, 0<ɛ<1, letf d (ɛ) denote the smallest integerf such that every range space of Vapnik-Chervonenkis dimensiond has anɛ-net of size at mostf. We solve a problem of Haussler and Welzl by showing that ifd≥2, then
$$d - 2 + \frac{2}{{d + 2}} \leqslant \mathop {\lim }\limits_{\varepsilon \to 0} \frac{{f_d (\varepsilon )}}{{(1/\varepsilon )\log (1/\varepsilon )}} \leqslant d.$$
Further, we prove thatf1(ɛ)=max(2, ⌌ 1/ɛ ⌍−1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.

Keywords

Maximal Element Discrete Comput Geom Computational Geometry Range Space Simple Polygon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • János Komlós
    • 1
  • János Pach
    • 2
    • 3
  • Gerhard Woeginger
    • 4
  1. 1.Department of MathematicsRutgers UniversityNew BrunswickUSA
  2. 2.Mathematical InstituteHungarian Academy of SciencesBudapestHungary
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  4. 4.Institut für MathematikTechnische Universität GrazGrazAustria

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