Document Open Access Logo

Clustering in Polygonal Domains

Authors Mark de Berg , Leyla Biabani, Morteza Monemizadeh, Leonidas Theocharous



PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2023.23.pdf
  • Filesize: 0.98 MB
  • 15 pages

Document Identifiers

Author Details

Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Leyla Biabani
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Morteza Monemizadeh
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Leonidas Theocharous
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Cite AsGet BibTex

Mark de Berg, Leyla Biabani, Morteza Monemizadeh, and Leonidas Theocharous. Clustering in Polygonal Domains. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 23:1-23:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.23

Abstract

We study various clustering problems for a set D of n points in a polygonal domain P under the geodesic distance. We start by studying the discrete k-median problem for D in P. We develop an exact algorithm which runs in time poly(n,m) + n^O(√k), where m is the complexity of the domain. Subsequently, we show that our approach can also be applied to solve the k-center problem with z outliers in the same running time. Next, we turn our attention to approximation algorithms. In particular, we study the k-center problem in a simple polygon and show how to obtain a (1+ε)-approximation algorithm which runs in time 2^{O((k log(k))/ε)} (n log(m) + m). To obtain this, we demonstrate that a previous approach by Bădoiu et al. [Bâdoiu et al., 2002; Bâdoiu and Clarkson, 2003] that works in ℝ^d, carries over to the setting of simple polygons. Finally, we study the 1-center problem in a simple polygon in the presence of z outliers. We show that a coreset C of size O(z) exists, such that the 1-center of C is a 3-approximation of the 1-center of D, when z outliers are allowed. This result is actually more general and carries over to any metric space, which to the best of our knowledge was not known so far. By extending this approach, we show that for the 1-center problem under the Euclidean metric in ℝ², there exists an ε-coreset of size O(z/ε).

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • clustering
  • geodesic distance
  • coreset
  • outliers

Metrics

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

References

  1. Pankaj K. Agarwal and Micha Sharir. Planar geometric location problems. Algorithmica, 11(2):185-195, 1994. URL: https://doi.org/10.1007/BF01182774.
  2. Hee-Kap Ahn, Luis Barba, Prosenjit Bose, Jean-Lou De Carufel, Matias Korman, and Eunjin Oh. A linear-time algorithm for the geodesic center of a simple polygon. Discret. Comput. Geom., 56(4):836-859, 2016. URL: https://doi.org/10.1007/s00454-016-9796-0.
  3. Henk Alkema, Mark de Berg, Morteza Monemizadeh, and Leonidas Theocharous. TSP in a Simple Polygon. 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 5:1-5:14, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2022.5.
  4. Mihai Bâdoiu and Kenneth L. Clarkson. Smaller core-sets for balls. In Proc. 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), pages 801-802, 2003. Google Scholar
  5. Mihai Bâdoiu, Sariel Har-Peled, and Piotr Indyk. Approximate clustering via core-sets. In Proc. 34th Annual ACM Symposium on Theory of Computing (STOC 2002), pages 250-257, 2002. URL: https://doi.org/10.1145/509907.509947.
  6. Matteo Ceccarello, Andrea Pietracaprina, and Geppino Pucci. Solving k-center clustering (with outliers) in mapreduce and streaming, almost as accurately as sequentially. Proc. VLDB Endow., 12(7):766-778, 2019. URL: https://doi.org/10.14778/3317315.3317319.
  7. Mark de Berg, Leyla Biabani, and Morteza Monemizadeh. k-center clustering with outliers in the MPC and streaming model. In Proc. 37th IEEE International Parallel and Distributed Processing Symposium (IPDPS 2023), pages 853-863, 2023. URL: https://doi.org/10.1109/IPDPS54959.2023.00090.
  8. Hristo Djidjev and Shankar M. Venkatesan. Reduced constants for simple cycle graph separation. Acta Informatica, 34(3):231-243, 1997. URL: https://doi.org/10.1007/s002360050082.
  9. Zvi Drezner. The p-centre problem-heuristic and optimal algorithms. The Journal of the Operational Research Society, 35(8):741-748, 1984. URL: http://www.jstor.org/stable/2581980.
  10. Yunjun Gao and Baihua Zheng. Continuous obstructed nearest neighbor queries in spatial databases. In Proc. ACM SIGMOD International Conference on Management of Data (SIGMOD 2009), pages 577-590, 2009. URL: https://doi.org/10.1145/1559845.1559906.
  11. Leonidas J. Guibas and John Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989. URL: https://doi.org/10.1016/0022-0000(89)90041-X.
  12. R. Z. Hwang, R. C. Chang, and Richard C. T. Lee. The searching over separators strategy to solve some NP-hard problems in subexponential time. Algorithmica, 9(4):398-423, 1993. URL: https://doi.org/10.1007/BF01228511.
  13. R. Z. Hwang, Richard C. T. Lee, and R. C. Chang. The slab dividing approach to solve the Euclidean p-center problem. Algorithmica, 9(1):1-22, 1993. URL: https://doi.org/10.1007/BF01185335.
  14. Dániel Marx and Michał Pilipczuk. Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams. ACM Trans. Algorithms, 18(2), 2022. URL: https://doi.org/10.1145/3483425.
  15. Nimrod Megiddo. Linear-time algorithms for linear programming in ℝ³ and related problems. In Proc. 23rd Annual Symposium on Foundations of Computer Science (FOCS 1982), pages 329-338, 1982. URL: https://doi.org/10.1109/SFCS.1982.24.
  16. Nimrod Megiddo and Kenneth J. Supowit. On the complexity of some common geometric location problems. SIAM Journal on Computing, 13(1):182-196, 1984. URL: https://doi.org/10.1137/0213014.
  17. Gary L. Miller. Finding small simple cycle separators for 2-connected planar graphs. Journal of Computer and System Sciences, 32(3):265-279, 1986. URL: https://doi.org/10.1016/0022-0000(86)90030-9.
  18. Eunjin Oh, Sang Won Bae, and Hee-Kap Ahn. Computing a geodesic two-center of points in a simple polygon. Computational Geometry, 82:45-59, 2019. URL: https://doi.org/10.1016/j.comgeo.2019.05.001.
  19. Michael Ian Shamos and Dan Hoey. Closest-point problems. In Proc. 16th Annual Symposium on Foundations of Computer Science (FOCS 1975), pages 151-162, 1975. URL: https://doi.org/10.1109/SFCS.1975.8.
  20. J. J. Sylvester. A question in the geometry of situation. Quarterly Journal of Pure and Applied Mathematics, 1857. Google Scholar
  21. A.K.H. Tung, J. Hou, and Jiawei Han. Spatial clustering in the presence of obstacles. In Proc. 17th International Conference on Data Engineering, pages 359-367, 2001. URL: https://doi.org/10.1109/ICDE.2001.914848.
  22. Haitao Wang. On the planar two-center problem and circular hulls. Discrete & Computational Geometry, 68(4):1175-1226, 2022. URL: https://doi.org/10.1007/s00454-021-00358-5.
  23. Xin Wang and Howard J. HHamilton. Clustering spatial data in the presence of obstacles. International Journal on Artificial Intelligence Tools, 14:177-198, 2005. URL: https://doi.org/10.1142/S0218213005002053.
  24. Chenyi Xia, David Hsu, and Anthony K. H. Tung. A fast filter for obstructed nearest neighbor queries. In Key Technologies for Data Management, pages 203-215, 2004. Google Scholar
  25. O.R. Zaiane and Chi-Hoon Lee. Clustering spatial data in the presence of obstacles: a density-based approach. In Proc. International Database Engineering and Applications Symposium, pages 214-223, 2002. URL: https://doi.org/10.1109/IDEAS.2002.1029674.
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