Computing envelopes in four dimensions with applications

Pankaj K. Agarwal, Boris Aronov, Micha Sharir

    Research output: Contribution to journalArticle

    Abstract

    Let F ℱe 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, for any query point q, can determine in O(log2 n) time the function(s) of ℱ that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several 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 randomized expected 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)
    Pages (from-to)1714-1732
    Number of pages19
    JournalSIAM Journal on Computing
    Volume26
    Issue number6
    StatePublished - Dec 1997

    Fingerprint

    Envelope
    Data structures
    Computing
    Computational geometry
    Lower Envelopes
    Query
    Data Structures
    Point Sets
    Ray Shooting
    Point Location
    Simple Polygon
    Computational Geometry
    Randomized Algorithms
    Ring or annulus
    Moving Objects
    Maximum Degree
    Preprocessing
    Nearest Neighbor
    Covering
    Face

    Keywords

    • Closest pair
    • Lower envelopes
    • Point location
    • Ray shooting

    ASJC Scopus subject areas

    • Computational Theory and Mathematics
    • Applied Mathematics
    • Theoretical Computer Science

    Cite this

    Agarwal, P. K., Aronov, B., & Sharir, M. (1997). Computing envelopes in four dimensions with applications. SIAM Journal on Computing, 26(6), 1714-1732.

    Computing envelopes in four dimensions with applications. / Agarwal, Pankaj K.; Aronov, Boris; Sharir, Micha.

    In: SIAM Journal on Computing, Vol. 26, No. 6, 12.1997, p. 1714-1732.

    Research output: Contribution to journalArticle

    Agarwal, PK, Aronov, B & Sharir, M 1997, 'Computing envelopes in four dimensions with applications', SIAM Journal on Computing, vol. 26, no. 6, pp. 1714-1732.
    Agarwal PK, Aronov B, Sharir M. Computing envelopes in four dimensions with applications. SIAM Journal on Computing. 1997 Dec;26(6):1714-1732.
    Agarwal, Pankaj K. ; Aronov, Boris ; Sharir, Micha. / Computing envelopes in four dimensions with applications. In: SIAM Journal on Computing. 1997 ; Vol. 26, No. 6. pp. 1714-1732.
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