Computing the distance between piecewise-linear bivariate functions

Guillaume Moroz, Boris Aronov

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

    Abstract

    We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L 2-norm, that is ||f - g|| 2 = √∫∫ M(f - g) 2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.

    Original languageEnglish (US)
    Title of host publicationProceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
    Pages288-293
    Number of pages6
    StatePublished - 2012
    Event23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan
    Duration: Jan 17 2012Jan 19 2012

    Other

    Other23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
    CountryJapan
    CityKyoto
    Period1/17/121/19/12

    Fingerprint

    Triangulation
    Piecewise Linear
    Computing
    Interpolation
    Linear Interpolation
    Polynomials
    Triangle
    Norm
    Polynomial
    Evaluation

    ASJC Scopus subject areas

    • Software
    • Mathematics(all)

    Cite this

    Moroz, G., & Aronov, B. (2012). Computing the distance between piecewise-linear bivariate functions. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 (pp. 288-293)

    Computing the distance between piecewise-linear bivariate functions. / Moroz, Guillaume; Aronov, Boris.

    Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. p. 288-293.

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

    Moroz, G & Aronov, B 2012, Computing the distance between piecewise-linear bivariate functions. in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. pp. 288-293, 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, 1/17/12.
    Moroz G, Aronov B. Computing the distance between piecewise-linear bivariate functions. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. p. 288-293
    Moroz, Guillaume ; Aronov, Boris. / Computing the distance between piecewise-linear bivariate functions. Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. pp. 288-293
    @inproceedings{9369d7b9aad14bd381593bae64dd2f83,
    title = "Computing the distance between piecewise-linear bivariate functions",
    abstract = "We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L 2-norm, that is ||f - g|| 2 = √∫∫ M(f - g) 2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious na{\"i}ve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.",
    author = "Guillaume Moroz and Boris Aronov",
    year = "2012",
    language = "English (US)",
    isbn = "9781611972108",
    pages = "288--293",
    booktitle = "Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012",

    }

    TY - GEN

    T1 - Computing the distance between piecewise-linear bivariate functions

    AU - Moroz, Guillaume

    AU - Aronov, Boris

    PY - 2012

    Y1 - 2012

    N2 - We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L 2-norm, that is ||f - g|| 2 = √∫∫ M(f - g) 2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.

    AB - We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L 2-norm, that is ||f - g|| 2 = √∫∫ M(f - g) 2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multi-point evaluation of a certain type of polynomials. We also present an application to terrain matching.

    UR - http://www.scopus.com/inward/record.url?scp=84860166197&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=84860166197&partnerID=8YFLogxK

    M3 - Conference contribution

    AN - SCOPUS:84860166197

    SN - 9781611972108

    SP - 288

    EP - 293

    BT - Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

    ER -