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Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D

Authors Lotte Blank , Jacobus Conradi , Anne Driemel , Benedikt Kolbe , André Nusser , Marena Richter

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  • Theory of computation → Computational geometry
Keywords
  • Fréchet distance under translation
  • Fréchet distance under scaling
  • time series
  • shape matching

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Abstract

The Fréchet distance is a computational mainstay for comparing polygonal curves. The Fréchet distance under translation, which is a translation invariant version, considers the similarity of two curves independent of their location in space. It is defined as the minimum Fréchet distance that arises from allowing arbitrary translations of the input curves. This problem and numerous variants of the Fréchet distance under some transformations have been studied, with more work concentrating on the discrete Fréchet distance, leaving a significant gap between the discrete and continuous versions of the Fréchet distance under transformations. Our contribution is twofold: First, we present an algorithm for the Fréchet distance under translation on 1-dimensional curves of complexity n with a running time of 𝒪(n^{8/3} log³ n). To achieve this, we develop a novel framework for the problem for 1-dimensional curves, which also applies to other scenarios and leads to our second contribution. We present an algorithm with the same running time of 𝒪(n^{8/3} log³ n) for the Fréchet distance under scaling for 1-dimensional curves. For both algorithms we match the running times of the discrete case and improve the previously best known bounds of 𝒪̃(n⁴). Our algorithms rely on technical insights but are conceptually simple, essentially reducing the continuous problem to the discrete case across different length scales.

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Lotte Blank, Jacobus Conradi, Anne Driemel, Benedikt Kolbe, André Nusser, and Marena Richter. Transforming Dogs on the Line: On the Fréchet Distance Under Translation or Scaling in 1D. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.22

Author Details

Lotte Blank
  • University of Bonn, Germany
Jacobus Conradi
  • University of Bonn, Germany
Anne Driemel
  • University of Bonn, Germany
Benedikt Kolbe
  • University of Bonn, Germany
André Nusser
  • Université Côte d'Azur, CNRS, Inria, Sophia Antipolis, France
Marena Richter
  • University of Bonn, Germany

Funding

  • Blank, Lotte: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 459420781 (FOR AlgoForGe).
  • Conradi, Jacobus: Funded by the iBehave Network: Sponsored by the Ministry of Culture and Science of the State of North Rhine-Westphalia.
  • Driemel, Anne: Affiliated with Lamarr Institute for Machine Learning and Artificial Intelligence.
  • Kolbe, Benedikt: This work was partially supported by the Lamarr Institute for Machine Learning and Artificial Intelligence.
  • 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.

References

  1. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. Internat. J. Comput. Geom. Appl., 5(1-2):75-91, 1995. URL: https://doi.org/10.1142/S0218195995000064.
  2. Helmut Alt, Christian Knauer, and Carola Wenk. Matching polygonal curves with respect to the Fréchet distance. In Afonso Ferreira and Horst Reichel, editors, STACS 2001, 18th Annual Symposium on Theoretical Aspects of Computer Science, Dresden, Germany, February 15-17, 2001, Proceedings, volume 2010 of Lecture Notes in Computer Science, pages 63-74. Springer, 2001. URL: https://doi.org/10.1007/3-540-44693-1_6.
  3. Helmut Alt, Christian Knauer, and Carola Wenk. Comparison of distance measures for planar curves. Algorithmica, 38(1):45-58, 2004. URL: https://doi.org/10.1007/S00453-003-1042-5.
  4. Rinat Ben Avraham, Haim Kaplan, and Micha Sharir. A faster algorithm for the discrete Fréchet distance under translation. CoRR, abs/1501.03724, 2015. URL: https://arxiv.org/abs/1501.03724.
  5. Abdullah Biran and Aleksandar Jeremic. Ecg bio-identification using Fréchet classifiers: A proposed methodology based on modeling the dynamic change of the ecg features. Biomedical Signal Processing and Control, 82:104575, 2023. URL: https://doi.org/10.1016/j.bspc.2023.104575.
  6. Lotte Blank and Anne Driemel. A faster algorithm for the Fréchet distance in 1d for the imbalanced case. In Timothy M. Chan, Johannes Fischer, John Iacono, and Grzegorz Herman, editors, 32nd Annual European Symposium on Algorithms, ESA 2024, September 2-4, 2024, Royal Holloway, London, United Kingdom, volume 308 of LIPIcs, pages 28:1-28:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ESA.2024.28.
  7. Sotiris Brakatsoulas, Dieter Pfoser, Randall Salas, and Carola Wenk. On map-matching vehicle tracking data. In Proc. 31st International Conf. Very Large Data Bases (VLDB'05), pages 853-864, 2005. URL: http://www.vldb.org/archives/website/2005/program/paper/fri/p853-brakatsoulas.pdf.
  8. Karl Bringmann. Why walking the dog takes time: Fréchet distance has no strongly subquadratic algorithms unless SETH fails. In Proc. 55th Ann. IEEE Symposium on Foundations of Computer Science (FOCS'14), pages 661-670, 2014. URL: https://doi.org/10.1109/FOCS.2014.76.
  9. Karl Bringmann, Anne Driemel, André Nusser, and Ioannis Psarros. Tight bounds for approximate near neighbor searching for time series under the Fréchet distance. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 517-550. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.25.
  10. Karl Bringmann, Marvin Künnemann, and André Nusser. When Lipschitz walks your dog: Algorithm engineering of the discrete Fréchet distance under translation. In 28th Annual European Symposium on Algorithms (ESA), volume 173 of LIPIcs, pages 25:1-25:17, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.25.
  11. Karl Bringmann, Marvin Künnemann, and André Nusser. Discrete Fréchet distance under translation: Conditional hardness and an improved algorithm. ACM Trans. Algorithms, 17(3):25:1-25:42, 2021. URL: https://doi.org/10.1145/3460656.
  12. Kevin Buchin, Maike Buchin, Wouter Meulemans, and Wolfgang Mulzer. Four soviets walk the dog - with an application to Alt’s conjecture. In Proc. 25th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA'14), pages 1399-1413, 2014. URL: https://doi.org/10.1137/1.9781611973402.103.
  13. Kevin Buchin, Anne Driemel, Natasja van de L'Isle, and André Nusser. klcluster: Center-based clustering of trajectories. In Farnoush Banaei Kashani, Goce Trajcevski, Ralf Hartmut Güting, Lars Kulik, and Shawn D. Newsam, editors, Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, SIGSPATIAL 2019, Chicago, IL, USA, November 5-8, 2019, pages 496-499. ACM, 2019. URL: https://doi.org/10.1145/3347146.3359111.
  14. Kevin Buchin, André Nusser, and Sampson Wong. Computing continuous dynamic time warping of time series in polynomial time. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 22:1-22:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.SOCG.2022.22.
  15. 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 Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 2887-2901. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.179.
  16. Siu-Wing Cheng and Haoqiang Huang. Fréchet distance in subquadratic time. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 5100-5113, 2025. URL: https://doi.org/10.1137/1.9781611978322.173.
  17. Ian R. Cleasby, Ewan D. Wakefield, Barbara J. Morrissey, Thomas W. Bodey, Steven C. Votier, Stuart Bearhop, and Keith C. Hamer. Using time-series similarity measures to compare animal movement trajectories in ecology. Behavioral Ecology and Sociobiology, 73(11):151, November 2019. URL: https://doi.org/10.1007/s00265-019-2761-1.
  18. Richard Cole. Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34(1):200-208, 1987. URL: https://doi.org/10.1145/7531.7537.
  19. Jacobus Conradi, Anne Driemel, and Benedikt Kolbe. ( 1 + ε ) -ANN Data Structure for Curves via Subspaces of Bounded Doubling Dimension. Computing in Geometry and Topology, 3(2):1-22, 2024. URL: https://doi.org/10.57717/cgt.v3i2.45.
  20. Anne Driemel and Sariel Har-Peled. Jaywalking your dog: Computing the Fréchet distance with shortcuts. SIAM J. Comput., 42(5):1830-1866, 2013. URL: https://doi.org/10.1137/120865112.
  21. Anne Driemel, Sariel Har-Peled, and Carola Wenk. Approximating the Fréchet distance for realistic curves in near linear time. In Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, SoCG '10, pages 365-374, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1810959.1811019.
  22. Anne Driemel, Amer Krivošija, and Christian Sohler. Clustering time series under the Fréchet distance. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 766-785. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH55.
  23. Alon Efrat, Piotr Indyk, and Suresh Venkatasubramanian. Pattern matching for sets of segments. In Proc. 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'01), pages 295-304, 2001. URL: http://dl.acm.org/citation.cfm?id=365411.365463.
  24. Omrit Filtser and Matthew J. Katz. Algorithms for the discrete Fréchet distance under translation. J. Comput. Geom., 11(1):156-175, 2020. URL: https://doi.org/10.20382/JOCG.V11I1A7.
  25. Omrit Filtser and Matthew J. Katz. Variants of the discrete Fréchet distance under translation. Journal of Computational Geometry, 11(1):156-175, 2020. Google Scholar
  26. M Maurice Fréchet. Sur quelques points du calcul fonctionnel. Rendiconti del Circolo Matematico di Palermo (1884-1940), 22(1):1-72, 1906. Google Scholar
  27. Sariel Har-Peled and Benjamin Raichel. The Fréchet distance revisited and extended. ACM Trans. Algorithms, 10(1):3:1-3:22, 2014. URL: https://doi.org/10.1145/2532646.
  28. Minghui Jiang, Ying Xu, and Binhai Zhu. Protein structure-structure alignment with discrete Fréchet distance. J. Bioinformatics and Computational Biology, 6(01):51-64, 2008. URL: https://doi.org/10.1142/S0219720008003278.
  29. Hongli Niu and Jun Wang. Volatility clustering and long memory of financial time series and financial price model. Digital Signal Processing, 23(2):489-498, 2013. URL: https://doi.org/10.1016/j.dsp.2012.11.004.
  30. Carola Wenk. Shape matching in higher dimensions. PhD thesis, Freie Universität Berlin, 2002. PhD Thesis. Google Scholar
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