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Computing a Subtrajectory Cluster from c-Packed Trajectories

Authors Joachim Gudmundsson , Zijin Huang , André van Renssen , Sampson Wong

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Subtrajectory cluster
  • c-packed trajectories
  • Computational geometry

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Abstract

We present a near-linear time approximation algorithm for the subtrajectory cluster problem of c-packed trajectories. Given a trajectory T of complexity n, an approximation factor ε, and a desired distance d, the problem involves finding m subtrajectories of T such that their pair-wise Fréchet distance is at most (1 + ε)d. At least one subtrajectory must be of length l or longer. A trajectory T is c-packed if the intersection of T and any ball B with radius r is at most c⋅r in length.
Previous results by Gudmundsson and Wong [Gudmundsson and Wong, 2022] established an Ω(n³) lower bound unless the Strong Exponential Time Hypothesis fails, and they presented an O(n³ log² n) time algorithm. We circumvent this conditional lower bound by studying subtrajectory cluster on c-packed trajectories, resulting in an algorithm with an O((c² n/ε²)log(c/ε)log(n/ε)) time complexity.

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Joachim Gudmundsson, Zijin Huang, André van Renssen, and Sampson Wong. Computing a Subtrajectory Cluster from c-Packed Trajectories. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ISAAC.2023.34

Author Details

Joachim Gudmundsson
  • The University of Sydney, Australia
Zijin Huang
  • The University of Sydney, Australia
André van Renssen
  • The University of Sydney, Australia
Sampson Wong
  • BARC, University of Copenhagen, Denmark

Funding

  • Gudmundsson, Joachim: Funded by the Australian Government through the Australian Research Council DP180102870.

Acknowledgements

The authors thank Kevin Buchin for the insightful discussion on the important Lemma 13. The authors thank the anonymous reviewers for their helpful feedback.

References

  1. Pankaj K. Agarwal, Kyle Fox, Kamesh Munagala, Abhinandan Nath, Jiangwei Pan, and Erin Taylor. Subtrajectory Clustering: Models and Algorithms. In Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS '18, pages 75-87, New York, NY, USA, May 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3196959.3196972.
  2. Sepideh Aghamolaei, Vahideh Keikha, Mohammad Ghodsi, and Ali Mohades. Windowing queries using Minkowski sum and their extension to MapReduce. The Journal of Supercomputing, 77(1):936-972, January 2021. URL: https://doi.org/10.1007/s11227-020-03299-7.
  3. Helmut Alt and Michael Godau. Computing the Fréchet distance between two polygonal curves. International Journal of Computational Geometry & Applications, 05:75-91, March 1995. URL: https://doi.org/10.1142/S0218195995000064.
  4. 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, October 2014. ISSN: 0272-5428. URL: https://doi.org/10.1109/FOCS.2014.76.
  5. Karl Bringmann and Marvin Künnemann. Improved approximation for Fréchet distance on c-packed curves matching conditional lower bounds. International Journal of Computational Geometry & Applications, 27:85-119, March 2017. URL: https://doi.org/10.1142/S0218195917600056.
  6. Frederik Brüning, Jacobus Conradi, and Anne Driemel. Faster Approximate Covering of Subcurves Under the Fréchet Distance. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms (ESA 2022), volume 244 of Leibniz International Proceedings in Informatics (LIPIcs), pages 28:1-28:16, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. ISSN: 1868-8969. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.28.
  7. 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, SIGSPATIAL '17, pages 1-10, New York, NY, USA, November 2017. Association for Computing Machinery. URL: https://doi.org/10.1145/3139958.3139964.
  8. Kevin Buchin, Maike Buchin, Joachim Gudmundsson, Maarten Löffler, and Jun Luo. Detecting commuting patterns by clustering subtrajectories. International Journal of Computational Geometry & Applications, 21(03):253-282, June 2011. URL: https://doi.org/10.1142/S0218195911003652.
  9. Kevin Buchin, Maike Buchin, and Yusu Wang. Exact algorithms for partial curve matching via the Fréchet distance. In Proceedings of the twentieth annual ACM-SIAM symposium on Discrete algorithms, SODA '09, pages 645-654, USA, January 2009. Society for Industrial and Applied Mathematics. Google Scholar
  10. Claudia Cavallaro, Armir Bujari, Luca Foschini, Giuseppe Di Modica, and Paolo Bellavista. Measuring the impact of COVID-19 restrictions on mobility: A real case study from Italy. Journal of Communications and Networks, 23(5):340-349, October 2021. URL: https://doi.org/10.23919/JCN.2021.000034.
  11. Cheng Chang and Baoyao Zhou. Multi-granularity visualization of trajectory clusters using sub-trajectory clustering. In 2009 IEEE International Conference on Data Mining Workshops, pages 577-582, December 2009. URL: https://doi.org/10.1109/ICDMW.2009.24.
  12. Daniel Chen, Anne Driemel, Leonidas J. Guibas, Andy Nguyen, and Carola Wenk. Approximate map matching with respect to the Fréchet distance. In 2011 Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), Proceedings, pages 75-83. Society for Industrial and Applied Mathematics, January 2011. URL: https://doi.org/10.1137/1.9781611972917.8.
  13. Jacobus Conradi and Anne Driemel. On Computing the k-Shortcut Fréchet Distance. In Mikobackslashlaj Bojańczyk, Emanuela Merelli, and David P. Woodruff, editors, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022), volume 229 of Leibniz International Proceedings in Informatics (LIPIcs), pages 46:1-46:20, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. ISSN: 1868-8969. URL: https://doi.org/10.4230/LIPIcs.ICALP.2022.46.
  14. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer, Berlin, Heidelberg, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  15. Anne Driemel, Sariel Har-Peled, and Carola Wenk. Approximating the Fréchet distance for realistic curves in near linear time. Discrete & Computational Geometry, 48(1):94-127, July 2012. URL: https://doi.org/10.1007/s00454-012-9402-z.
  16. Alexander Gavruskin, Bakhadyr Khoussainov, Mikhail Kokho, and Jiamou Liu. Dynamic algorithms for monotonic interval scheduling problem. Theoretical Computer Science, 562:227-242, January 2015. URL: https://doi.org/10.1016/j.tcs.2014.09.046.
  17. Joachim Gudmundsson, Martin P. Seybold, and John Pfeifer. On practical nearest sub-trajectory queries under the Fréchet distance. In Proceedings of the 29th International Conference on Advances in Geographic Information Systems, SIGSPATIAL '21, pages 596-605, New York, NY, USA, November 2021. Association for Computing Machinery. URL: https://doi.org/10.1145/3474717.3484264.
  18. Joachim Gudmundsson, Martin P. Seybold, and Sampson Wong. Map matching queries on realistic input graphs under the Fréchet distance. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1464-1492. Society for Industrial and Applied Mathematics, January 2023. URL: https://doi.org/10.1137/1.9781611977554.ch53.
  19. Joachim Gudmundsson, Yuan Sha, and Sampson Wong. Approximating the packedness of polygonal curves. Computational Geometry, 108:101920, January 2023. URL: https://doi.org/10.1016/j.comgeo.2022.101920.
  20. Joachim Gudmundsson and Michiel Smid. Fréchet queries in geometric trees. In Hans L. Bodlaender and Giuseppe F. Italiano, editors, Algorithms – ESA 2013, pages 565-576, Berlin, Heidelberg, 2013. Springer. URL: https://doi.org/10.1007/978-3-642-40450-4_48.
  21. Joachim Gudmundsson, Andreas Thom, and Jan Vahrenhold. Of motifs and goals: mining trajectory data. In Proceedings of the 20th International Conference on Advances in Geographic Information Systems, SIGSPATIAL '12, pages 129-138, New York, NY, USA, November 2012. Association for Computing Machinery. URL: https://doi.org/10.1145/2424321.2424339.
  22. Joachim Gudmundsson and Nacho Valladares. A GPU approach to subtrajectory clustering using the Fréchet distance. IEEE Transactions on Parallel and Distributed Systems, 26(4):924-937, April 2015. URL: https://doi.org/10.1109/TPDS.2014.2317713.
  23. Joachim Gudmundsson and Thomas Wolle. Football analysis using spatio-temporal tools. In Proceedings of the 20th International Conference on Advances in Geographic Information Systems, SIGSPATIAL '12, pages 566-569, New York, NY, USA, November 2012. Association for Computing Machinery. URL: https://doi.org/10.1145/2424321.2424417.
  24. Joachim Gudmundsson and Sampson Wong. Cubic upper and lower bounds for subtrajectory clustering under the continuous Fréchet distance. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 173-189. Society for Industrial and Applied Mathematics, January 2022. URL: https://doi.org/10.1137/1.9781611977073.9.
  25. Amin Hosseinpoor Milaghardan, Rahim Ali Abbaspour, Christophe Claramunt, and Alireza Chehreghan. An activity-based framework for detecting human movement patterns in an urban environment. Transactions in GIS, 25(4):1825-1848, 2021. URL: https://doi.org/10.1111/tgis.12749.
  26. Jae-Gil Lee, Jiawei Han, and Kyu-Young Whang. Trajectory clustering: a partition-and-group framework. In Proceedings of the 2007 ACM SIGMOD international conference on Management of data, SIGMOD '07, pages 593-604, New York, NY, USA, June 2007. Association for Computing Machinery. URL: https://doi.org/10.1145/1247480.1247546.
  27. Anil Maheshwari, Jörg-Rüdiger Sack, Kaveh Shahbaz, and Hamid Zarrabi-Zadeh. Fréchet distance with speed limits. Computational Geometry, 44(2):110-120, February 2011. URL: https://doi.org/10.1016/j.comgeo.2010.09.008.
  28. Daniel D. Sleator and Robert Endre Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26(3):362-391, June 1983. URL: https://doi.org/10.1016/0022-0000(83)90006-5.
  29. Panagiotis Tampakis, Nikos Pelekis, Christos Doulkeridis, and Yannis Theodoridis. Scalable distributed subtrajectory clustering. In Proceedings of the 2019 IEEE International Conference on Big Data (Big Data), pages 950-959. IEEE Computer Society, December 2019. URL: https://doi.org/10.1109/BigData47090.2019.9005563.
  30. Zheng Wang, Cheng Long, and Gao Cong. Similar sports play retrieval with deep reinforcement learning. IEEE Transactions on Knowledge and Data Engineering, pages 1-1, 2021. URL: https://doi.org/10.1109/TKDE.2021.3136881.
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