Computing envelopes in four dimensions with applications

Pankaj K. Agarwal, Boris Aronov, Micha Sharir

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

    Abstract

    Let F be a collection of n d-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(nd+ε), for any ε > 0. For d = 3, by combining this algorithm with the point location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n3+ε), for any ε > 0, a data structure of size O(n3+ε) that, given any query point q, can determine in O(log2 n) time whether q lies above, below or on the envelope. As a consequence, we obtain improved algorithmic solutions to many problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the biggest stick in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in time O(n17/11+ε), for any ε > 0, improving previous solutions that run in time O(n8/5+ε). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane, and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n3+ε) storage and preprocessing time, for any ε > 0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.

    Original languageEnglish (US)
    Title of host publicationProceedings of the Annual Symposium on Computational Geometry
    PublisherPubl by ACM
    Pages348-358
    Number of pages11
    ISBN (Print)0897916484, 9780897916486
    DOIs
    StatePublished - 1994
    EventProceedings of the 10th Annual Symposium on Computational Geometry - Stony Brook, NY, USA
    Duration: Jun 6 1994Jun 8 1994

    Publication series

    NameProceedings of the Annual Symposium on Computational Geometry

    Other

    OtherProceedings of the 10th Annual Symposium on Computational Geometry
    CityStony Brook, NY, USA
    Period6/6/946/8/94

    ASJC Scopus subject areas

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

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  • Cite this

    Agarwal, P. K., Aronov, B., & Sharir, M. (1994). Computing envelopes in four dimensions with applications. In Proceedings of the Annual Symposium on Computational Geometry (pp. 348-358). (Proceedings of the Annual Symposium on Computational Geometry). Publ by ACM. https://doi.org/10.1145/177424.178081