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

Boris Aronov, Esther Ezra, Micha Sharir

    Research output: Contribution to journalArticle

    Abstract

    We show the existence of ε-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 and "fat" triangular ranges and for point sets in R3 and axis-parallel boxes; these are the first known nontrivial 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 ε-nets of size O (1/ε 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 or of Even, Rawitz, and Shahar, we obtain improved approximation factors (computable in expected polynomial time by a randomized algorithm) for the HITTING SET or the SET COVER problems associated with the corresponding range spaces.

    Original languageEnglish (US)
    Pages (from-to)3248-3282
    Number of pages35
    JournalSIAM Journal on Computing
    Volume39
    Issue number7
    DOIs
    StatePublished - Sep 1 2010

    Keywords

    • Exponential decay lemma
    • Geometric range spaces
    • HITTING SET
    • SET COVER
    • ε-nets

    ASJC Scopus subject areas

    • Computer Science(all)
    • Mathematics(all)

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