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Faster Fréchet Distance Under Transformations

Authors Kevin Buchin , Maike Buchin , Zijin Huang , André Nusser , Sampson Wong

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Fréchet distance
  • curve similarity
  • shape matching

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Abstract

We study the problem of computing the Fréchet distance between two polygonal curves under transformations. First, we consider translations in the Euclidean plane. Given two curves π and σ of total complexity n and a threshold δ ≥ 0, we present an 𝒪̃(n^{7 + 1/3}) time algorithm to determine whether there exists a translation t ∈ ℝ² such that the Fréchet distance between π and σ + t is at most δ. This improves on the previous best result, which is an 𝒪(n⁸) time algorithm.
We then generalize this result to any class of rationally parameterized transformations, which includes translation, rotation, scaling, and arbitrary affine transformations. For a class T of rationally parametrized transformations with k degrees of freedom, we show that one can determine whether there is a transformation τ ∈ T such that the Fréchet distance between π and τ(σ) is at most δ in 𝒪̃(n^{3k+4/3}) time.

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Kevin Buchin, Maike Buchin, Zijin Huang, André Nusser, and Sampson Wong. Faster Fréchet Distance Under Transformations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.36

Author Details

Kevin Buchin
  • Technical University of Dortmund, Germany
Maike Buchin
  • Ruhr University Bochum, Germany
Zijin Huang
  • University of Sydney, Australia
André Nusser
  • Université Côte d'Azur, CNRS, Inria, Sophia Antipolis, France
Sampson Wong
  • BARC, University of Copenhagen, Denmark

Funding

  • Nusser, André: This work was supported by the French government through the France 2030 investment plan managed by the National Research Agency (ANR), as part of the Initiative of Excellence of Université Côte d'Azur under reference number ANR-15-IDEX-01.
  • Wong, Sampson: This work was supported by the European Union under a Marie Skłodowska-Curie Actions (MSCA) Postdoctoral Fellowship Grant number 101146276.

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