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A Subquadratic Algorithm for Computing the L₁-Distance Between Two Terrains

Authors Pankaj K. Agarwal , Boris Aronov , Olivier Devillers , Christian Knauer , Guillaume Moroz

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Terrain similarity
  • volume computation
  • polynomial interpolation
  • geometric cuttings
  • bivariate multipoint evaluation

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Abstract

We study the problem of computing the L₁-distance between two piecewise-linear bivariate functions f and g, defined over a bounded polygonal domain 𝕄 ⊂ ℝ², that is, computing the quantity ‖f-g‖₁ = ∫_𝕄 |f(x,y)-g(x,y)| dx dy. If f and g are defined by linear interpolation over triangulations 𝐓_f and 𝐓_g, respectively, of 𝕄 with a total of n triangles, we show that ‖f-g‖₁ can be computed in Õ(n^α) time, where α = max{(ω+1)/2, 8/5}, ω is the matrix multiplication exponent, and Õ notation hides factors of the form n^ε for any ε > 0. This bound holds for the currently best known value of ω, which is approximately 2.37. More generally, if the complexity of the overlay of 𝐓_f and 𝐓_g is κ, then the runtime of our algorithm is Õ(κ^{α-1}n^{2-α}).

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Pankaj K. Agarwal, Boris Aronov, Olivier Devillers, Christian Knauer, and Guillaume Moroz. A Subquadratic Algorithm for Computing the L₁-Distance Between Two Terrains. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.4

Author Details

Pankaj K. Agarwal
  • Department of Computer Science, Duke University, Durham, NC, USA
Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Olivier Devillers
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Christian Knauer
  • University of Bayreuth, Germany
Guillaume Moroz
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

Funding

G.M. acknowledges the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-24-CE48-1899 (project StratMESH). Work by P.K.A. has been partially supported by NSF grants CCF-20-07556, CCF-22-23870, and IIS-24-02823, and by the Binational Science Foundation Grant 2022131. Work by B.A. has been partially supported by NSF grants CCF-15-40656 and CCF-20-08551.

Acknowledgements

This work was initiated at the Workshop on Problems in Computational Geometry at Bellairs Research Institute, Holetown, Barbados in 2012, and the authors wish to thank Éric Colin de Verdière for useful discussions. The authors also wish to thank Bruno Salvy for useful remarks on recent multipoint evaluation results.

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