Distance-sensitive planar point location

Boris Aronov, Mark De Berg, David Eppstein, Marcel Roeloffzen, Bettina Speckmann

    Research output: Contribution to journalArticle

    Abstract

    Let S be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γi that the query point lies in a polygon Pi of S. We show how to preprocess S such that the query time for a point p∈Pi depends on γi and, in addition, on the distance from p to the boundary of Pi - the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time O(min (logn, 1+log area(Pi)/γiΔp2)), where Δp is the shortest Euclidean distance of the query point p to the boundary of Pi. Our structure uses O(n) space and O(nlogn) preprocessing time. It is based on a decomposition of the regions of S into convex quadrilaterals and triangles with the following property: for any point p ∈ Pi, the quadrilateral or triangle containing p has area Ω(Δp2). For the special case where S is a subdivision of the unit square and γi=area(Pi), we present a simpler solution that achieves a query time of O(min(logn,log 1/Δp2)). The latter solution can be extended to convex subdivisions in three dimensions.

    Original languageEnglish (US)
    Pages (from-to)17-31
    Number of pages15
    JournalComputational Geometry: Theory and Applications
    Volume54
    DOIs
    StatePublished - Apr 1 2016

    Fingerprint

    Point Location
    Pi
    Query
    Subdivision
    Decomposition
    Triangle
    Euclidean Distance
    Polygon
    Preprocessing
    Three-dimension
    Decompose
    Unit

    Keywords

    • Mesh generation
    • Point location
    • Quadtree

    ASJC Scopus subject areas

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

    Cite this

    Aronov, B., De Berg, M., Eppstein, D., Roeloffzen, M., & Speckmann, B. (2016). Distance-sensitive planar point location. Computational Geometry: Theory and Applications, 54, 17-31. https://doi.org/10.1016/j.comgeo.2016.02.001

    Distance-sensitive planar point location. / Aronov, Boris; De Berg, Mark; Eppstein, David; Roeloffzen, Marcel; Speckmann, Bettina.

    In: Computational Geometry: Theory and Applications, Vol. 54, 01.04.2016, p. 17-31.

    Research output: Contribution to journalArticle

    Aronov, B, De Berg, M, Eppstein, D, Roeloffzen, M & Speckmann, B 2016, 'Distance-sensitive planar point location', Computational Geometry: Theory and Applications, vol. 54, pp. 17-31. https://doi.org/10.1016/j.comgeo.2016.02.001
    Aronov, Boris ; De Berg, Mark ; Eppstein, David ; Roeloffzen, Marcel ; Speckmann, Bettina. / Distance-sensitive planar point location. In: Computational Geometry: Theory and Applications. 2016 ; Vol. 54. pp. 17-31.
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