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Faster, Deterministic and Space Efficient Subtrajectory Clustering

Authors Ivor van der Hoog , Thijs van der Horst , Tim Ophelders

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Fréchet distance
  • clustering
  • set cover

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Abstract

Given a trajectory T and a distance Δ, we wish to find a set C of curves of complexity at most 𝓁, such that we can cover T with subcurves that each are within Fréchet distance Δ to at least one curve in C. We call C an (𝓁,Δ)-clustering and aim to find an (𝓁,Δ)-clustering of minimum cardinality. This problem variant was introduced by Akitaya et al. (2021) and shown to be NP-complete. The main focus has therefore been on bicriteria approximation algorithms, allowing for the clustering to be an (𝓁, Θ(Δ))-clustering of roughly optimal size.
We present algorithms that construct (𝓁,4Δ)-clusterings of 𝒪(k log n) size, where k is the size of the optimal (𝓁, Δ)-clustering. We use 𝒪(n³) space and 𝒪(k n³ log⁴ n) time. Our algorithms significantly improve upon the clustering quality (improving the approximation factor in Δ) and size (whenever 𝓁 ∈ Ω(log n / log k)). We offer deterministic running times improving known expected bounds by a factor near-linear in 𝓁. Additionally, we match the space usage of prior work, and improve it substantially, by a factor super-linear in n𝓁, when compared to deterministic results.

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Ivor van der Hoog, Thijs van der Horst, and Tim Ophelders. Faster, Deterministic and Space Efficient Subtrajectory Clustering. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 133:1-133:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.133

Author Details

Ivor van der Hoog
  • Department of Applied Mathematics and Computer Science, Technical University of Denmark, Lyngby, Denmark
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

  • van der Hoog, Ivor: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 899987, and from the Carlsberg Foundation via Eva Rotenberg’s Young Researcher Fellowship CF21-0302 "Graph Algorithms with Geometric Applications".
  • Ophelders, Tim: Partially supported by the Dutch Research Council (NWO) under the project number VI.Veni.212.260.

Acknowledgements

We thank Jacobus Conradi for pointing out an error in a preprint of this paper.

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