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A Near-Linear Time Exact Algorithm for the L₁-Geodesic Fréchet Distance Between Two Curves on the Boundary of a Simple Polygon

Authors Thijs van der Horst , Marc van Kreveld , Tim Ophelders , Bettina Speckmann

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  • Theory of computation → Computational geometry
Keywords
  • Fréchet distance
  • geodesic
  • simple polygon

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Abstract

Let P be a polygon with k vertices. Let R and B be two simple, interior disjoint curves on the boundary of P, with n and m vertices. We show how to compute the Fréchet distance between R and B using the geodesic L₁-distance in P in 𝒪(k log nm + (n+m) (log² nm log k + log⁴ nm)) time.

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Thijs van der Horst, Marc van Kreveld, Tim Ophelders, and Bettina Speckmann. A Near-Linear Time Exact Algorithm for the L₁-Geodesic Fréchet Distance Between Two Curves on the Boundary of a Simple Polygon. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 37:1-37:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.37

Author Details

Thijs van der Horst
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Marc van Kreveld
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
Tim Ophelders
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Bettina Speckmann
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

  • Ophelders, Tim: partially supported by the Dutch Research Council (NWO) under project no. VI.Veni.212.260.

References

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  2. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 5:75-91, 1995. URL: https://doi.org/10.1142/S0218195995000064.
  3. Karl Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails. In Proc. 55th Annual Symposium on Foundations of Computer Science (FOCS), pages 661-670, 2014. URL: https://doi.org/10.1109/FOCS.2014.76.
  4. Karl Bringmann and Marvin Künnemann. Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds. International Journal of Computational Geometry & Applications, 27(1-2):85-120, 2017. URL: https://doi.org/10.1142/S0218195917600056.
  5. Kevin Buchin, Tim Ophelders, and Bettina Speckmann. SETH says: Weak Fréchet distance is faster, but only if it is continuous and in one dimension. In Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2887-2901, 2019. URL: https://doi.org/10.1137/1.9781611975482.179.
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  7. Anne Driemel, Sariel Har-Peled, and Carola Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discrete & Computational Geometry, 48(1):94-127, 2012. URL: https://doi.org/10.1007/s00454-012-9402-z.
  8. Greg N. Frederickson and Donald B. Johnson. Generalized selection and ranking: Sorted matrices. SIAM Journal of Computing, 13(1):14-30, 1984. URL: https://doi.org/10.1137/0213002.
  9. Atlas F. Cook IV and Carola Wenk. Geodesic Fréchet distance inside a simple polygon. ACM Transactions on Algorithms, 7(1):9:1-9:19, 2010. URL: https://doi.org/10.1145/1868237.1868247.
  10. Andranik Mirzaian and Eshrat Arjomandi. Selection in X+Y and matrices with sorted rows and columns. Information Processing Letters, 20(1):13-17, 1985. URL: https://doi.org/10.1016/0020-0190(85)90123-1.
  11. Thijs van der Horst, Marc van Kreveld, Tim Ophelders, and Bettina Speckmann. The geodesic Fréchet distance between two curves bounding a simple polygon. CoRR, abs/2501.03834, 2025. URL: https://doi.org/10.48550/arXiv.2501.03834.
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