Document Open Access Logo

Data Structures for Approximate Fréchet Distance for Realistic Curves

Authors Ivor van der Hoog , Eva Rotenberg , Sampson Wong

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2024.56.pdf
  • Filesize: 0.9 MB
  • 18 pages

Document Identifiers

Author Details

Ivor van der Hoog
  • DTU Compute, Technical University of Denmark, Lyngby, Denmark
Eva Rotenberg
  • DTU Compute, Technical University of Denmark, Lyngby, Denmark
Sampson Wong
  • Department of Computer Science, University of Copenhagen, Denmark

Funding

  • van der Hoog, Ivor: Supported by Independent Research Fund Denmark grant 2020-2023 (9131-00044B) "Dynamic Network Analysis".
  • Rotenberg, Eva: Partially supported by Independent Research Fund Denmark grant 2020-2023 (9131-00044B) "Dynamic Network Analysis" and the Carlsberg Young Researcher Fellowship CF21-0302 "Graph Algorithms with Geometric Applications".

Cite As Get BibTex

Ivor van der Hoog, Eva Rotenberg, and Sampson Wong. Data Structures for Approximate Fréchet Distance for Realistic Curves. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.56

Abstract

The Fréchet distance is a popular distance measure between curves P and Q. Conditional lower bounds prohibit (1+ε)-approximate Fréchet distance computations in strongly subquadratic time, even when preprocessing P using any polynomial amount of time and space. As a consequence, the Fréchet distance has been studied under realistic input assumptions, for example, assuming both curves are c-packed.
In this paper, we study c-packed curves in Euclidean space ℝ^d and in general geodesic metrics 𝒳. In ℝ^d, we provide a nearly-linear time static algorithm for computing the (1+ε)-approximate continuous Fréchet distance between c-packed curves. Our algorithm has a linear dependence on the dimension d, as opposed to previous algorithms which have an exponential dependence on d.
In general geodesic metric spaces X, little was previously known. We provide the first data structure, and thereby the first algorithm, under this model. Given a c-packed input curve P with n vertices, we preprocess it in O(n log n) time, so that given a query containing a constant ε and a curve Q with m vertices, we can return a (1+ε)-approximation of the discrete Fréchet distance between P and Q in time polylogarithmic in n and linear in m, 1/ε, and the realism parameter c.
Finally, we show several extensions to our data structure; to support dynamic extend/truncate updates on P, to answer map matching queries, and to answer Hausdorff distance queries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
Keywords
  • Fréchet distance
  • data structures
  • approximation algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Pankaj K Agarwal, Rinat Ben Avraham, Haim Kaplan, and Micha Sharir. Computing the discrete fréchet distance in subquadratic time. SIAM Journal on Computing, 43(2):429-449, 2014. URL: https://doi.org/10.1137/130920526.
  2. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 5(01n02):75-91, 1995. URL: https://doi.org/10.1142/S0218195995000064.
  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. Boris Aronov, Omrit Filtser, Michael Horton, Matthew J. Katz, and Khadijeh Sheikhan. Efficient nearest-neighbor query and clustering of planar curves. In Zachary Friggstad, Jörg-Rüdiger Sack, and Mohammad R Salavatipour, editors, Algorithms and Data Structures, pages 28-42, Cham, 2019. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-24766-9_3.
  5. Boris Aronov, Sariel Har-Peled, Christian Knauer, Yusu Wang, and Carola Wenk. Fréchet distance for curves, revisited. In Yossi Azar and Thomas Erlebach, editors, Algorithms - ESA 2006, 14th Annual European Symposium, Zurich, Switzerland, September 11-13, 2006, Proceedings, volume 4168 of Lecture Notes in Computer Science, pages 52-63. Springer, 2006. URL: https://doi.org/10.1007/11841036_8.
  6. Alessandro Bombelli, Lluis Soler, Eric Trumbauer, and Kenneth D Mease. Strategic air traffic planning with Fréchet distance aggregation and rerouting. Journal of Guidance, Control, and Dynamics, 40(5):1117-1129, 2017. Google Scholar
  7. Sotiris Brakatsoulas, Dieter Pfoser, Randall Salas, and Carola Wenk. On map-matching vehicle tracking data. In Proceedings of the 31st international conference on Very large data bases, 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 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 661-670. IEEE, 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 and Marvin Künnemann. Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds. International Journal of Computational Geometry & Applications, 27(01n02):85-119, 2017. URL: https://doi.org/10.1142/S0218195917600056.
  11. Kevin Buchin, Maike Buchin, David Duran, Brittany Terese Fasy, Roel Jacobs, Vera Sacristan, Rodrigo I Silveira, Frank Staals, and Carola Wenk. Clustering trajectories for map construction. In Proceedings of the 25th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pages 1-10, 2017. URL: https://doi.org/10.1145/3139958.3139964.
  12. Kevin Buchin, Maike Buchin, Wouter Meulemans, and Wolfgang Mulzer. Four Soviets walk the dog: Improved bounds for computing the Fréchet distance. Discrete & Computational Geometry, 58(1):180-216, 2017. URL: https://doi.org/10.1007/S00454-017-9878-7.
  13. 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 Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2887-2901. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.179.
  14. Maike Buchin, Bernhard Kilgus, and Andrea Kölzsch. Group diagrams for representing trajectories. International Journal of Geographical Information Science, 34(12):2401-2433, 2020. URL: https://doi.org/10.1080/13658816.2019.1684498.
  15. Maike Buchin, Ivor van der Hoog, Tim Ophelders, Lena Schlipf, Rodrigo I Silveira, and Frank Staals. Efficient Fréchet distance queries for segments. European Symposium on Algorithms, 2022. Google Scholar
  16. Daniel Chen, Anne Driemel, Leonidas J. Guibas, Andy Nguyen, and Carola Wenk. Approximate map matching with respect to the fréchet distance. In Matthias Müller-Hannemann and Renato Fonseca F. Werneck, editors, Proceedings of the Thirteenth Workshop on Algorithm Engineering and Experiments, ALENEX 2011, Holiday Inn San Francisco Golden Gateway, San Francisco, California, USA, January 22, 2011, pages 75-83. SIAM, 2011. URL: https://doi.org/10.1137/1.9781611972917.8.
  17. Jacobus Conradi, Anne Driemel, and Benedikt Kolbe. Revisiting the fr$$1'echet distance between piecewise smooth curves. arXiv preprint arXiv:2401.03339, 2024. URL: https://doi.org/10.48550/arXiv.2401.03339.
  18. Mark De Berg, Atlas F Cook IV, and Joachim Gudmundsson. Fast Fréchet queries. Computational Geometry, 46(6):747-755, 2013. URL: https://doi.org/10.1016/J.COMGEO.2012.11.006.
  19. Mark de Berg, Ali D Mehrabi, and Tim Ophelders. Data structures for Fréchet queries in trajectory data. In 29th Canadian Conference on Computational Geometry (CCCG'17), pages 214-219, 2017. Google Scholar
  20. Thomas Devogele. A new merging process for data integration based on the discrete Fréchet distance. In Advances in spatial data handling, pages 167-181. Springer, 2002. Google Scholar
  21. Anne Driemel and Sariel Har-Peled. Jaywalking your dog: computing the Fréchet distance with shortcuts. SIAM Journal on Computing, 42(5):1830-1866, 2013. URL: https://doi.org/10.1137/120865112.
  22. Anne Driemel, Sariel Har-Peled, and Carola Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discret. Comput. Geom., 48(1):94-127, 2012. URL: https://doi.org/10.1007/s00454-012-9402-z.
  23. Anne Driemel and Ioannis Psarros. (2+ε)-ANN for time series under the Fréchet distance. Workshop on Algorithms and Data structures (WADS), 2021. Google Scholar
  24. Anne Driemel, Ioannis Psarros, and Melanie Schmidt. Sublinear data structures for short Fréchet queries. CoRR, abs/1907.04420, 2019. URL: https://arxiv.org/abs/1907.04420.
  25. Anne Driemel, Ivor van der Hoog, and Eva Rotenberg. On the discrete Fréchet distance in a graph. In International Symposium on Computational Geometry (SoCG 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  26. Thomas Eiter and Heikki Mannila. Computing discrete Fréchet distance. Technical Report CD-TR 94/64, Christian Doppler Laboratory for Expert Systems, TU Vienna, Austria, 1994. Google Scholar
  27. Arnold Filtser and Omrit Filtser. Static and streaming data structures for fréchet distance queries. In Dániel Marx, editor, Symposium on Discrete Algorithms (SODA) 2021, pages 1150-1170. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.71.
  28. Arnold Filtser, Omrit Filtser, and Matthew J. Katz. Approximate nearest neighbor for curves - simple, efficient, and deterministic. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), volume 168 of Leibniz International Proceedings in Informatics (LIPIcs), pages 48:1-48:19, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.48.
  29. Omrit Filtser. Universal approximate simplification under the discrete fréchet distance. Inf. Process. Lett., 132:22-27, 2018. URL: https://doi.org/10.1016/j.ipl.2017.10.002.
  30. Joachim Gudmundsson, Martin P. Seybold, and Sampson Wong. Map matching queries on realistic input graphs under the fréchet distance. Symposium on Discrete Algorithms (SODA), 2023. Google Scholar
  31. Joachim Gudmundsson, André van Renssen, Zeinab Saeidi, and Sampson Wong. Fréchet distance queries in trajectory data. In The Third Iranian Conference on Computational Geometry (ICCG 2020), pages 29-32, 2020. Google Scholar
  32. Leonidas J Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. In Symposium on Computational geometry (SoCG), 1987. Google Scholar
  33. Sariel Har-Peled. Geometric approximation algorithms, volume 173 of Mathematical Surveys and Monographs. American Mathematical Soc., 2011. Google Scholar
  34. Minghui Jiang, Ying Xu, and Binhai Zhu. Protein structure-structure alignment with discrete Fréchet distance. Journal of bioinformatics and computational biology, 6(01):51-64, 2008. URL: https://doi.org/10.1142/S0219720008003278.
  35. Maximilian Konzack, Thomas McKetterick, Tim Ophelders, Maike Buchin, Luca Giuggioli, Jed Long, Trisalyn Nelson, Michel A Westenberg, and Kevin Buchin. Visual analytics of delays and interaction in movement data. International Journal of Geographical Information Science, 31(2):320-345, 2017. URL: https://doi.org/10.1080/13658816.2016.1199806.
  36. Yaowei Long and Seth Pettie. Planar distance oracles with better time-space tradeoffs. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021, pages 2517-2537. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.149.
  37. Ariane Mascret, Thomas Devogele, Iwan Le Berre, and Alain Hénaff. Coastline matching process based on the discrete Fréchet distance. In Progress in Spatial Data Handling, pages 383-400. Springer, 2006. Google Scholar
  38. Nimrod Megiddo. Applying parallel computation algorithms in the design of serial algorithms. J. ACM, 30(4):852-865, 1983. URL: https://doi.org/10.1145/2157.322410.
  39. Otfried Schwarzkopf and Jules Vleugels. Range searching in low-density environments. Inf. Process. Lett., 60(3):121-127, 1996. URL: https://doi.org/10.1016/S0020-0190(96)00154-8.
  40. Roniel S. De Sousa, Azzedine Boukerche, and Antonio A. F. Loureiro. Vehicle trajectory similarity: Models, methods, and applications. ACM Comput. Surv., 53(5), September 2020. URL: https://doi.org/10.1145/3406096.
  41. E Sriraghavendra, K Karthik, and Chiranjib Bhattacharyya. Fréchet distance based approach for searching online handwritten documents. In Ninth International Conference on Document Analysis and Recognition (ICDAR 2007), volume 1, pages 461-465. IEEE, 2007. URL: https://doi.org/10.1109/ICDAR.2007.4378752.
  42. Han Su, Shuncheng Liu, Bolong Zheng, Xiaofang Zhou, and Kai Zheng. A survey of trajectory distance measures and performance evaluation. The VLDB Journal, 29(1):3-32, 2020. URL: https://doi.org/10.1007/S00778-019-00574-9.
  43. Mikkel Thorup. Compact oracles for reachability and approximate distances in planar digraphs. Journal of the ACM (JACM), 51(6):993-1024, 2004. URL: https://doi.org/10.1145/1039488.1039493.
  44. Ivor van der Hoog, Eva Rotenberg, and Sampson Wong. Data structures for approximate discrete Fréchet distance. CoRR, abs/2212.07124, 2022. URL: https://doi.org/10.48550/arXiv.2212.07124.
  45. A. Frank van der Stappen. Motion planning amidst fat obstacles. University Utrecht, 1994. Google Scholar
  46. Rene Van Oostrum and Remco Veltkamp. Parametric search made practical. In Symposium on Computational Geometry (C), pages 1-9, 2002. Google Scholar
  47. Dong Xie, Feifei Li, and Jeff M Phillips. Distributed trajectory similarity search. Proceedings of the VLDB Endowment, 10(11):1478-1489, 2017. URL: https://doi.org/10.14778/3137628.3137655.
  48. Daming Xu. Well-separated pair decompositions for doubling metric spaces. PhD thesis, Carleton University, 2005. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact