Document Open Access Logo

Clustering Point Sets Revisited

Authors Md. Billal Hossain , Benjamin Raichel

PDF HTML 5

Files

Thumbnail PDF
PDF
LIPIcs.WADS.2025.38.pdf
  • Filesize: 0.82 MB
  • 16 pages
HTML5 Logo HTML (experimental)
LIPIcs.WADS.2025.38.html

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Clustering
  • k-center
  • k-median
  • k-means

Metrics

Abstract

In the sets clustering problem one is given a collection of point sets 𝒫 = {P_1,… P_m} in ℝ^d, where for any set of k centers in ℝ^d, each P_i is assigned to its nearest center as determine by some local cost functions. The goal is then to select a set of k centers to minimize some global cost function of the corresponding local assignment costs. Specifically, we consider either summing or taking the maximum cost over all P_i, where for each P_i the cost of assigning it to a center c is either max_{p ∈ P_i} ‖c-p‖, ∑_{p ∈ P_i} ‖c-p‖, or ∑_{p ∈ P_i} ‖c-p‖².
Different combinations of the global and local cost functions naturally generalize the k-center, k-median, and k-means clustering problems. In this paper, we improve the prior results for the natural generalization of k-center, give the first result for the natural generalization of k-means, and give results for generalizations of k-median and k-center which differ from those previously studied.

Cite As Get BibTex

Md. Billal Hossain and Benjamin Raichel. Clustering Point Sets Revisited. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 38:1-38:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.38

Author Details

Md. Billal Hossain
  • Department of Computer Science, University of Texas at Dallas, TX, USA
Benjamin Raichel
  • Department of Computer Science, University of Texas at Dallas, TX, USA

Funding

Work on this paper was partially supported by NSF CAREER AF Award CCF-1750780 and NSF AF Award CCF-2311179.

References

  1. Pankaj K. Agarwal and Cecilia Magdalena Procopiuc. Exact and approximation algorithms for clustering. Algorithmica, 33(2):201-226, 2002. URL: https://doi.org/10.1007/s00453-001-0110-y.
  2. Mihai Badoiu and Kenneth L. Clarkson. Smaller core-sets for balls. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 801-802. ACM/SIAM, 2003. URL: http://dl.acm.org/citation.cfm?id=644108.644240.
  3. Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. J. Comput. Syst. Sci., 68(2):417-441, 2004. URL: https://doi.org/10.1016/J.JCSS.2003.07.014.
  4. Ke Chen. On coresets for k-median and k-means clustering in metric and euclidean spaces and their applications. SIAM J. Comput., 39(3):923-947, 2009. URL: https://doi.org/10.1137/070699007.
  5. Vincent Cohen-Addad, Karthik C. S., and Euiwoong Lee. Johnson coverage hypothesis: Inapproximability of k-means and k-median in lp-metrics. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1493-1530. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.63.
  6. Eduard Eiben, Fedor V. Fomin, Petr A. Golovach, William Lochet, Fahad Panolan, and Kirill Simonov. EPTAS for k-means clustering of affine subspaces. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2649-2659. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.157.
  7. Tomás Feder and Daniel H. Greene. Optimal algorithms for approximate clustering. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC), pages 434-444. ACM, 1988. URL: https://doi.org/10.1145/62212.62255.
  8. Dan Feldman, Morteza Monemizadeh, and Christian Sohler. A PTAS for k-means clustering based on weak coresets. In Proceedings of the 23rd ACM Symposium on Computational Geometry (SOCG), pages 11-18. ACM, 2007. URL: https://doi.org/10.1145/1247069.1247072.
  9. Jie Gao, Michael Langberg, and Leonard J. Schulman. Clustering lines in high-dimensional space: Classification of incomplete data. ACM Trans. Algorithms, 7(1), December 2010. URL: https://doi.org/10.1145/1868237.1868246.
  10. T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985. URL: https://doi.org/10.1016/0304-3975(85)90224-5.
  11. Sariel Har-Peled and Soham Mazumdar. On coresets for k-means and k-median clustering. In László Babai, editor, Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pages 291-300. ACM, 2004. URL: https://doi.org/10.1145/1007352.1007400.
  12. D. S. Hochbaum and D. B. Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10(2):180-184, 1985. URL: https://doi.org/10.1287/moor.10.2.180.
  13. W. Hsu and G. L. Nemhauser. Easy and hard bottleneck location problems. Discrete Applied Mathematics, 1(3):209-215, 1979. URL: https://doi.org/10.1016/0166-218X(79)90044-1.
  14. Hongyao Huang, Georgiy Klimenko, and Benjamin Raichel. Clustering with neighborhoods. In 32nd International Symposium on Algorithms and Computation (ISAAC), volume 212 of LIPIcs, pages 6:1-6:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ISAAC.2021.6.
  15. Ibrahim Jubran, Murad Tukan, Alaa Maalouf, and Dan Feldman. Sets clustering. In Proceedings of the 37th International Conference on Machine Learning, (ICML), volume 119 of Proceedings of Machine Learning Research, pages 4994-5005. PMLR, 2020. URL: http://proceedings.mlr.press/v119/jubran20a.html.
  16. Tapas Kanungo, David M. Mount, Nathan S. Netanyahu, Christine D. Piatko, Ruth Silverman, and Angela Y. Wu. A local search approximation algorithm for k-means clustering. Comput. Geom., 28(2-3):89-112, 2004. URL: https://doi.org/10.1016/J.COMGEO.2004.03.003.
  17. Amit Kumar, Yogish Sabharwal, and Sandeep Sen. Linear-time approximation schemes for clustering problems in any dimensions. J. ACM, 57(2), February 2010. URL: https://doi.org/10.1145/1667053.1667054.
  18. Sagi Lotan, Ernesto Evgeniy Sanches Shayda, and Dan Feldman. Coreset for line-sets clustering. In Advances in Neural Information Processing Systems (NeurIPS), 2022. URL: http://papers.nips.cc/paper_files/paper/2022/hash/f2ce95887c34393af4eb240d60017860-Abstract-Conference.html.
  19. Qingzhou Wang and Kam-Hoi Cheng. A heuristic algorithm for the k-center problem with vertex weight. In International Symposium on Algorithms (SIGAL), volume 450 of Lecture Notes in Computer Science, pages 388-396. Springer, 1990. URL: https://doi.org/10.1007/3-540-52921-7_88.
  20. Guang Xu and Jinhui Xu. Efficient approximation algorithms for clustering point-sets. Computational Geometry, 43(1):59-66, 2010. Special Issue on the 14th Annual Fall Workshop. URL: https://doi.org/10.1016/j.comgeo.2007.12.002.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact