Computing the distance between piecewise-linear bivariate functions

Guillaume Moroz, Boris Aronov

    Research output: Contribution to journalArticlepeer-review


    We consider the problem of computing the distance between two piecewise-linear bivariate functions f and g defined over a common domain M, induced by the L2 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 and g is defined over another such triangulation, the obvious naive algorithm requires Θ (n2) arithmetic operations to compute this distance.We show that it is possible to compute it in O(nlog4 nlog log n) arithmetic operations by reducing the problem to multipoint evaluation of a certain type of polynomials. We also present several generalizations and an application to terrain matching.

    Original languageEnglish (US)
    Article number3
    JournalACM Transactions on Algorithms
    Issue number1
    StatePublished - Feb 2016


    • Multipoint evaluation
    • Piecewise-linear function
    • Polyhedral terrain

    ASJC Scopus subject areas

    • Mathematics (miscellaneous)


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