Small-size ε-nets for axis-parallel rectangles and boxes

Boris Aronov, Esther Ezra, Micha Sharir

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    We show the existence of e-nets of size O (1/ε log log 1/ε) for planar point sets and axis-parallel rectangular ranges. The same bound holds for points in the plane with "fat" triangular ranges, and for point sets in R3 and axis-parallel boxes; these are the first known non-trivial bounds for these range spaces. Our technique also yields improved bounds on the size of ε-nets in the more general context considered by Clarkson and Varadarajan. For example, we show the existence of e-nets of size O (1/εe log log log 1/ε) for the dual range space of "fat" regions and planar point sets (where the regions are the ground objects and the ranges are subsets stabbed by points). Plugging our bounds into the technique of Brönnimann and Goodrich, we obtain improved approximation factors (computable in randomized polynomial time) for the hitting set or the set cover problems associated with the corresponding range spaces.

    Original languageEnglish (US)
    Title of host publicationSTOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing
    Pages639-648
    Number of pages10
    DOIs
    StatePublished - 2009
    Event41st Annual ACM Symposium on Theory of Computing, STOC '09 - Bethesda, MD, United States
    Duration: May 31 2009Jun 2 2009

    Other

    Other41st Annual ACM Symposium on Theory of Computing, STOC '09
    CountryUnited States
    CityBethesda, MD
    Period5/31/096/2/09

    Fingerprint

    Oils and fats
    Polynomials

    Keywords

    • E-nets
    • Geometric range spaces
    • Hitting set
    • Randomized algorithms
    • Set cover

    ASJC Scopus subject areas

    • Software

    Cite this

    Aronov, B., Ezra, E., & Sharir, M. (2009). Small-size ε-nets for axis-parallel rectangles and boxes. In STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing (pp. 639-648) https://doi.org/10.1145/1536414.1536501

    Small-size ε-nets for axis-parallel rectangles and boxes. / Aronov, Boris; Ezra, Esther; Sharir, Micha.

    STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing. 2009. p. 639-648.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Aronov, B, Ezra, E & Sharir, M 2009, Small-size ε-nets for axis-parallel rectangles and boxes. in STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing. pp. 639-648, 41st Annual ACM Symposium on Theory of Computing, STOC '09, Bethesda, MD, United States, 5/31/09. https://doi.org/10.1145/1536414.1536501
    Aronov B, Ezra E, Sharir M. Small-size ε-nets for axis-parallel rectangles and boxes. In STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing. 2009. p. 639-648 https://doi.org/10.1145/1536414.1536501
    Aronov, Boris ; Ezra, Esther ; Sharir, Micha. / Small-size ε-nets for axis-parallel rectangles and boxes. STOC'09 - Proceedings of the 2009 ACM International Symposium on Theory of Computing. 2009. pp. 639-648
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