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Computing Optimal Epsilon-Nets Is as Easy as Finding an Unhit Set

Author Nabil H. Mustafa

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Author Details

Nabil H. Mustafa
  • Université Paris-Est, Laboratoire d'Informatique Gaspard-Monge, ESIEE Paris, France

Funding

This research was supported by the grant ANR SAGA (JCJC-14-CE25-0016-01).

Acknowledgements

We thank the reviewers for insightful feedback that helped the content and presentation of this work.

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Nabil H. Mustafa. Computing Optimal Epsilon-Nets Is as Easy as Finding an Unhit Set. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 87:1-87:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.87

Abstract

Given a set system (X, R) with VC-dimension d, the celebrated result of Haussler and Welzl (1987) showed that there exists an epsilon-net for (X, R) of size O(d/epsilon log 1/epsilon). Furthermore, the algorithm is simple: just take a uniform random sample from X! However, for many geometric set systems this bound is sub-optimal and since then, there has been much work presenting improved bounds and algorithms tailored to specific geometric set systems.
In this paper, we consider the following natural algorithm to compute an epsilon-net: start with an initial random sample N. Iteratively, as long as N is not an epsilon-net for R, pick any unhit set S in R (say, given by an Oracle), and add O(1) randomly chosen points from S to N.
We prove that the above algorithm computes, in expectation, epsilon-nets of asymptotically optimal size for all known cases of geometric set systems. Furthermore, it makes O(1/epsilon) calls to the Oracle. In particular, this implies that computing optimal-sized epsilon-nets are as easy as computing an unhit set in the given set system.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
Keywords
  • epsilon-nets
  • Geometric Set Systems

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