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
    Editors Anon
    PublisherPubl by ACM
    Pages348-358
    Number of pages11
    ISBN (Print)0897916484
    StatePublished - 1994
    EventProceedings of the 10th Annual Symposium on Computational Geometry - Stony Brook, NY, USA
    Duration: Jun 6 1994Jun 8 1994

    Other

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

    Fingerprint

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

    ASJC Scopus subject areas

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

    Cite this

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

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

    Proceedings of the Annual Symposium on Computational Geometry. ed. / Anon. Publ by ACM, 1994. p. 348-358.

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

    Agarwal, PK, Aronov, B & Sharir, M 1994, Computing envelopes in four dimensions with applications. in Anon (ed.), Proceedings of the Annual Symposium on Computational Geometry. Publ by ACM, pp. 348-358, Proceedings of the 10th Annual Symposium on Computational Geometry, Stony Brook, NY, USA, 6/6/94.
    Agarwal PK, Aronov B, Sharir M. Computing envelopes in four dimensions with applications. In Anon, editor, Proceedings of the Annual Symposium on Computational Geometry. Publ by ACM. 1994. p. 348-358
    Agarwal, Pankaj K. ; Aronov, Boris ; Sharir, Micha. / Computing envelopes in four dimensions with applications. Proceedings of the Annual Symposium on Computational Geometry. editor / Anon. Publ by ACM, 1994. pp. 348-358
    @inproceedings{7f8506c9d79143058fbdd5dd17a133ff,
    title = "Computing envelopes in four dimensions with applications",
    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.",
    author = "Agarwal, {Pankaj K.} and Boris Aronov and Micha Sharir",
    year = "1994",
    language = "English (US)",
    isbn = "0897916484",
    pages = "348--358",
    editor = "Anon",
    booktitle = "Proceedings of the Annual Symposium on Computational Geometry",
    publisher = "Publ by ACM",

    }

    TY - GEN

    T1 - Computing envelopes in four dimensions with applications

    AU - Agarwal, Pankaj K.

    AU - Aronov, Boris

    AU - Sharir, Micha

    PY - 1994

    Y1 - 1994

    N2 - 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.

    AB - 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.

    UR - http://www.scopus.com/inward/record.url?scp=0028124028&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0028124028&partnerID=8YFLogxK

    M3 - Conference contribution

    SN - 0897916484

    SP - 348

    EP - 358

    BT - Proceedings of the Annual Symposium on Computational Geometry

    A2 - Anon, null

    PB - Publ by ACM

    ER -