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k-Median Clustering Under Discrete Fréchet and Hausdorff Distances

Authors Abhinandan Nath, Erin Taylor

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Author Details

Abhinandan Nath
  • Mentor Graphics, Fremont, CA, USA
Erin Taylor
  • Duke University, Durham, NC, USA

Funding

  • Taylor, Erin: Work on this paper was supported by NSF under grants CCF-15-13816, CCF-15-46392, IIS-14-08846, and by an ARO grant W911NF-15-1-0408.

Acknowledgements

The authors would like to thank Pankaj K. Agarwal, Kamesh Munagala, and anonymous reviewers for helpful discussions and feedback.

Cite AsGet BibTex

Abhinandan Nath and Erin Taylor. k-Median Clustering Under Discrete Fréchet and Hausdorff Distances. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.58

Abstract

We give the first near-linear time (1+ε)-approximation algorithm for k-median clustering of polygonal trajectories under the discrete Fréchet distance, and the first polynomial time (1+ε)-approximation algorithm for k-median clustering of finite point sets under the Hausdorff distance, provided the cluster centers, ambient dimension, and k are bounded by a constant. The main technique is a general framework for solving clustering problems where the cluster centers are restricted to come from a simpler metric space. We precisely characterize conditions on the simpler metric space of the cluster centers that allow faster (1+ε)-approximations for the k-median problem. We also show that the k-median problem under Hausdorff distance is NP-Hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Clustering
  • k-median
  • trajectories
  • point sets
  • discrete Fréchet distance
  • Hausdorff distance

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References

  1. Marcel R. Ackermann, Johannes Blömer, and Christian Sohler. Clustering for metric and nonmetric distance measures. ACM Trans. Alg., 6(4):59:1-59:26, 2010. Google Scholar
  2. Pankaj K Agarwal and Cecilia M Procopiuc. Exact and approximation algorithms for clustering. Algorithmica, 33(2):201-226, 2002. Google Scholar
  3. Sanjeev Arora, Prabhakar Raghavan, and Satish Rao. Approximation schemes for Euclidean k-medians and related problems. In Proc. ACM Symp. Th. Computing, volume 98, pages 106-113, 1998. Google Scholar
  4. Arturs Backurs and Anastasios Sidiropoulos. Constant-distortion embeddings of hausdorff metrics into constant-dimensional l_p spaces. In APPROX/RANDOM. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  5. Mihai Bādoiu, Sariel Har-Peled, and Piotr Indyk. Approximate clustering via core-sets. In Proc. ACM Symp. Th. Computing, pages 250-257. ACM, 2002. Google Scholar
  6. Nicolas Basalto, Roberto Bellotti, Francesco De Carlo, Paolo Facchi, Ester Pantaleo, and Saverio Pascazio. Hausdorff clustering of financial time series. Physica A: Statistical Mechanics Applications, 379(2):635-644, 2007. Google Scholar
  7. Sergey Bereg, Minghui Jiang, Wencheng Wang, Boting Yang, and Binhai Zhu. Simplifying 3D polygonal chains under the discrete Fréchet distance. In Lat. Amer. Symp. Theoret. Informatics, pages 630-641. Springer, 2008. Google Scholar
  8. P. C. Besse, B. Guillouet, J. Loubes, and F. Royer. Review and perspective for distance-based clustering of vehicle trajectories. IEEE Tran. Intell. Transportation Sys., 17(11):3306-3317, 2016. Google Scholar
  9. Kevin Buchin, Anne Driemel, Joachim Gudmundsson, Michael Horton, Irina Kostitsyna, Maarten Löffler, and Martijn Struijs. Approximating (k,l)-center clustering for curves. In Proc. ACM-SIAM Symp. Disc. Alg., pages 2922-2938. SIAM, 2019. Google Scholar
  10. Kevin Buchin, Anne Driemel, and Martijn Struijs. On the hardness of computing an average curve. arXiv preprint, 2019. URL: http://arxiv.org/abs/1902.08053.
  11. Jinyang Chen, Rangding Wang, Liangxu Liu, and Jiatao Song. Clustering of trajectories based on hausdorff distance. In Proc. Int. Conf. Electronics Comm. Control, pages 1940-1944. IEEE, 2011. Google Scholar
  12. Vincent Cohen-Addad, Philip N Klein, and Claire Mathieu. Local search yields approximation schemes for k-means and k-median in euclidean and minor-free metrics. SIAM J. Computing, 48(2):644-667, 2019. Google Scholar
  13. Anne Driemel, Amer Krivošija, and Christian Sohler. Clustering time series under the Fréchet distance. In Proc. ACM-SIAM Symp. Disc. Alg., pages 766-785. SIAM, 2016. Google Scholar
  14. Thomas Eiter and Heikki Mannila. Computing discrete Fréchet distance. Technical report, Information Systems Dept., Technical University of Vienna, 1994. Google Scholar
  15. Martin Ester, Hans-Peter Kriegel, Jörg Sander, and Xiaowei Xu. A density-based algorithm for discovering clusters in large spatial databases with noise. In Proc. ACM Int. Conf. Know. Disc. Data Mining, volume 96, pages 226-231, 1996. Google Scholar
  16. Tomás Feder and Daniel Greene. Optimal algorithms for approximate clustering. In Proc. ACM Symp. Th. Computing, pages 434-444. ACM, 1988. Google Scholar
  17. Scott Gaffney and Padhraic Smyth. Trajectory clustering with mixtures of regression models. In Proc. ACM Int. Conf. Know. Disc. Data Mining, volume 99, pages 63-72, 1999. Google Scholar
  18. Venkatesan Guruswami and Piotr Indyk. Embeddings and non-approximability of geometric problems. In Proc. ACM-SIAM Symp. Disc. Alg., volume 3, pages 537-538, 2003. Google Scholar
  19. Sariel Har-Peled and Akash Kushal. Smaller coresets for k-median and k-means clustering. Disc. Computat. Geom., 37(1):3-19, 2007. Google Scholar
  20. Sariel Har-Peled and Soham Mazumdar. On coresets for k-means and k-median clustering. In Proc. ACM Symp. Th. Computing, pages 291-300. ACM, 2004. Google Scholar
  21. Felix Hausdorff. Grundzuge der mengenlehre, volume 61. American Mathematical Society, 1978. Google Scholar
  22. Chih-Chieh Hung, Wen-Chih Peng, and Wang-Chien Lee. Clustering and aggregating clues of trajectories for mining trajectory patterns and routes. Int. J. Very Large Databases, 24(2):169-192, 2015. Google Scholar
  23. Daniel P Huttenlocher, Gregory A Klanderman, and William J Rucklidge. Comparing images using the hausdorff distance. IEEE Trans. Pattern Anal. Machine Intell., 15(9):850-863, 1993. Google Scholar
  24. Kamal Jain, Mohammad Mahdian, and Amin Saberi. A new greedy approach for facility location problems. In Proc. ACM Symp. Th. Computing, pages 731-740. ACM, 2002. Google Scholar
  25. Stavros G Kolliopoulos and Satish Rao. A nearly linear-time approximation scheme for the euclidean k-median problem. SIAM J. Computing, 37(3):757-782, 2007. Google Scholar
  26. Amit Kumar, Yogish Sabharwal, and Sandeep Sen. Linear time algorithms for clustering problems in any dimensions. In Int. Coll. Automata Lang. Programming, pages 1374-1385. Springer, 2005. Google Scholar
  27. Amit Kumar, Yogish Sabharwal, and Sandeep Sen. Linear-time approximation schemes for clustering problems in any dimensions. J. ACM, 57(2):5, 2010. Google Scholar
  28. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. SIAM J. Computing, 45(2):530-547, 2016. Google Scholar
  29. Nimrod Megiddo and Kenneth J Supowit. On the complexity of some common geometric location problems. SIAM J. Computing, 13(1):182-196, 1984. Google Scholar
  30. Abhinandan Nath and Erin Taylor. k-Median clustering under discrete Fréchet and Hausdorff distances. CoRR, 2020. URL: http://arxiv.org/abs/2004.00722.
  31. Lin Qu, Fan Zhou, and YW Chen. Trajectory classification based on hausdorff distance for visual surveillance system. J. Jilin University, 6:1618-1624, 2009. Google Scholar
  32. Cynthia Sung, Dan Feldman, and Daniela Rus. Trajectory clustering for motion prediction. In IEEE/RSJ Int. Conf. Intell. Robots Syst., pages 1547-1552. IEEE, 2012. Google Scholar
  33. Hongteng Xu, Yang Zhou, Weiyao Lin, and Hongyuan Zha. Unsupervised trajectory clustering via adaptive multi-kernel-based shrinkage. In Proc. IEEE Int. Conf. Comp. Vision, pages 4328-4336, 2015. Google Scholar
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