Weak ε-nets have basis of size O (1 /ε log (1 / ε)) in any dimension

Nabil H. Mustafa, Saurabh Ray

    Research output: Contribution to journalArticle

    Abstract

    Given a set P of n points in ℝ d and >0, we consider the problem of constructing weak -nets for P. We show the following: pick a random sample Q of size O(1/εlog(1/ε)) from P. Then, with constant probability, a weak ε-net of P can be constructed from only the points of Q. This shows that weak -nets in ℝ d can be computed from a subset of P of size O(1/εlog(1/ε)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε. However, our final weak -nets still have a large size (with the dimension appearing in the exponent of 1/).

    Original languageEnglish (US)
    Pages (from-to)84-91
    Number of pages8
    JournalComputational Geometry: Theory and Applications
    Volume40
    Issue number1
    DOIs
    StatePublished - May 1 2008

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    Keywords

    • Combinatorial geometry
    • Hitting convex sets
    • Weak -nets

    ASJC Scopus subject areas

    • Geometry and Topology
    • Computer Science Applications
    • Control and Optimization
    • Computational Theory and Mathematics
    • Computational Mathematics

    Cite this

    Weak ε-nets have basis of size O (1 /ε log (1 / ε)) in any dimension. / Mustafa, Nabil H.; Ray, Saurabh.

    In: Computational Geometry: Theory and Applications, Vol. 40, No. 1, 01.05.2008, p. 84-91.

    Research output: Contribution to journalArticle

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