On approximate halfspace range counting and relative epsilon-approximations

Boris Aronov, Sariel Har-Peled, Micha Sharir

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

    Abstract

    The paper consists of two major parts. In the first part, we re-examine relative ε-approximations, previously studied in [12, 13, 18, 25], and their relation to certain geometric problems, most notably to approximate range counting. Wegive a simple constructive proof of their existence in general range spaces with finite VC dimension, and of a sharp bound on their size, close to the best known one. We then give a construction of smaller-size relative ε-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure-spanning trees with small relative crossing number, which we believe to be of independent interest. In the second part, we consider the approximate halfspace range-counting problem in Rd with relative error ε, and show that relative ε-approximations, combined with the shallow partitioning data structures of Matouŝek, yields ef- ficient solutions to this problem. For example, one of our data structures requires linear storage and O(n1+δ) preprocessingtime, for any > 0, and answers a query in time O(ε-n1-1/bd/2c2b log n), for any > 2/bd/2c; the choice of and affects b and the implied constants. Several variants and extensions are also discussed.

    Original languageEnglish (US)
    Title of host publicationProceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07
    Pages327-336
    Number of pages10
    DOIs
    StatePublished - 2007
    Event23rd Annual Symposium on Computational Geometry, SCG'07 - Gyeongju, Korea, Republic of
    Duration: Jun 6 2007Jun 8 2007

    Other

    Other23rd Annual Symposium on Computational Geometry, SCG'07
    CountryKorea, Republic of
    CityGyeongju
    Period6/6/076/8/07

    Fingerprint

    Half-space
    Data structures
    Counting
    Approximation
    Range of data
    Data Structures
    VC Dimension
    Counting Problems
    Crossing number
    Sharp Bound
    Relative Error
    Efficient Solution
    Spanning tree
    Higher Dimensions
    Partitioning
    Two Dimensions
    Query

    Keywords

    • Approximate range queries
    • Discrepancy
    • Epsilon-approximations
    • Halfspaces
    • Partition trees
    • Queries
    • Range
    • Range spaces
    • VC-dimension

    ASJC Scopus subject areas

    • Software
    • Geometry and Topology
    • Safety, Risk, Reliability and Quality
    • Chemical Health and Safety

    Cite this

    Aronov, B., Har-Peled, S., & Sharir, M. (2007). On approximate halfspace range counting and relative epsilon-approximations. In Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07 (pp. 327-336) https://doi.org/10.1145/1247069.1247128

    On approximate halfspace range counting and relative epsilon-approximations. / Aronov, Boris; Har-Peled, Sariel; Sharir, Micha.

    Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07. 2007. p. 327-336.

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

    Aronov, B, Har-Peled, S & Sharir, M 2007, On approximate halfspace range counting and relative epsilon-approximations. in Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07. pp. 327-336, 23rd Annual Symposium on Computational Geometry, SCG'07, Gyeongju, Korea, Republic of, 6/6/07. https://doi.org/10.1145/1247069.1247128
    Aronov B, Har-Peled S, Sharir M. On approximate halfspace range counting and relative epsilon-approximations. In Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07. 2007. p. 327-336 https://doi.org/10.1145/1247069.1247128
    Aronov, Boris ; Har-Peled, Sariel ; Sharir, Micha. / On approximate halfspace range counting and relative epsilon-approximations. Proceedings of the Twenty-third Annual Symposium on Computational Geometry, SCG'07. 2007. pp. 327-336
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