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Faster Fréchet Distance Approximation Through Truncated Smoothing

Authors Thijs van der Horst, Tim Ophelders

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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
Tim Ophelders
  • Department of Information and Computing Sciences, Utrecht University, The Netherlands
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

  • Ophelders, Tim: Partially supported by the Dutch Research Council (NWO) under the project number VI.Veni.212.260.

Cite AsGet BibTex

Thijs van der Horst and Tim Ophelders. Faster Fréchet Distance Approximation Through Truncated Smoothing. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.63

Abstract

The Fréchet distance is a popular distance measure for curves. Computing the Fréchet distance between two polygonal curves of n vertices takes roughly quadratic time, and conditional lower bounds suggest that even approximating to within a factor 3 cannot be done in strongly-subquadratic time, even in one dimension. The current best approximation algorithms present trade-offs between approximation quality and running time. Recently, van der Horst et al. (SODA, 2023) presented an O((n²/α) log³ n) time α-approximate algorithm for curves in arbitrary dimensions, for any α ∈ [1, n]. Our main contribution is an approximation algorithm for curves in one dimension, with a significantly faster running time of O(n log³ n + (n²/α³) log²n log log n). Additionally, we give an algorithm for curves in arbitrary dimensions that improves upon the state-of-the-art running time by a logarithmic factor, to O((n²/α) log² n). Both of our algorithms rely on a linear-time simplification procedure that in one dimension reduces the complexity of the reachable free space to O(n²/α) without making sacrifices in the asymptotic approximation factor.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Frécht distance
  • approximation algorithms
  • simplification

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References

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