Document Open Access Logo

Linear-Time (1+ε)-Approximation Algorithms for Two-Line-Center Problems

Authors Chaeyoon Chung , Anil Maheshwari , Michiel Smid

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2026.31.pdf
  • Filesize: 1.24 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Approximation algorithm
  • two-line-center problem
  • k-line-center problem
  • projective clustering
  • ε-certificate
  • ε-coreset
  • width of a point set

Metrics

Abstract

Given a set S of n points in the plane, we study the two-line-center problem: finding two lines that minimize the maximum distance from each point in S to its closest line. We present a (1+ε)-approximation algorithm for the two-line-center problem that runs in O((n/ε) log (1/ε)) time, which improves the previously best O(nlog n + (n/ε²) log (1/ε) + (1/ε³)log (1/ε))-time algorithm. We also consider three variants of this problem, in which the orientations of the two lines are restricted: (1) the orientation of one of the two lines is fixed, (2) the orientations of both lines are fixed, and (3) the two lines are required to be parallel. For each of these three variants, we give the first (1+ε)-approximation algorithm that runs in linear time. In particular, for the variant where the orientation of one of the two lines is fixed, we also give an improved exact algorithm that runs in O(n log n) time and show that it is optimal.

Cite As Get BibTex

Chaeyoon Chung, Anil Maheshwari, and Michiel Smid. Linear-Time (1+ε)-Approximation Algorithms for Two-Line-Center Problems. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.31

Author Details

Chaeyoon Chung
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Anil Maheshwari
  • School of Computer Science, Carleton University, Ottawa, Canada
Michiel Smid
  • School of Computer Science, Carleton University, Ottawa, Canada

Funding

Chaeyoon Chung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education(RS-2025-25436016). Anil Maheshwari and Michiel Smid were supported by NSERC.

Acknowledgements

This work was done while Chaeyoon Chung was visiting Carleton University.

References

  1. Pankaj K. Agarwal and Cecilia Magdalena Procopiuc. Approximation algorithms for projective clustering. J. Algorithms, 46(2):115-139, 2003. URL: https://doi.org/10.1016/S0196-6774(02)00295-X.
  2. Pankaj K. Agarwal, Cecilia Magdalena Procopiuc, and Kasturi R. Varadarajan. A (1+ε)-approximation algorithm for 2-line-center. Comput. Geom., 26(2):119-128, 2003. URL: https://doi.org/10.1016/S0925-7721(03)00017-8.
  3. Pankaj K. Agarwal, Cecilia Magdalena Procopiuc, and Kasturi R. Varadarajan. Approximation algorithms for a k-line center. Algorithmica, 42(3-4):221-230, 2005. URL: https://doi.org/10.1007/S00453-005-1166-X.
  4. Charu C. Aggarwal, Cecilia Magdalena Procopiuc, Joel L. Wolf, Philip S. Yu, and Jong Soo Park. Fast algorithms for projected clustering. In SIGMOD 1999, Proceedings ACM SIGMOD International Conference on Management of Data, June 1-3, 1999, Philadelphia, Pennsylvania, USA, pages 61-72. ACM Press, 1999. URL: https://doi.org/10.1145/304182.304188.
  5. Charu C. Aggarwal and Philip S. Yu. Finding generalized projected clusters in high dimensional spaces. In Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data, May 16-18, 2000, Dallas, Texas, USA, pages 70-81. ACM, 2000. URL: https://doi.org/10.1145/342009.335383.
  6. Taehoon Ahn and Sang Won Bae. Constrained two-line center problems. In 35th International Symposium on Algorithms and Computation, ISAAC 2024, Sydney, Australia, December 8-11, 2024, volume 322 of LIPIcs, pages 5:1-5:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ISAAC.2024.5.
  7. Mihai Badoiu, Sariel Har-Peled, and Piotr Indyk. Approximate clustering via core-sets. In Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 250-257. ACM, 2002. URL: https://doi.org/10.1145/509907.509947.
  8. Sang Won Bae. Minimum-width double-strip and parallelogram annulus. Theor. Comput. Sci., 833:133-146, 2020. URL: https://doi.org/10.1016/J.TCS.2020.05.045.
  9. Gill Barequet and Sariel Har-Peled. Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms, 38(1):91-109, 2001. URL: https://doi.org/10.1006/JAGM.2000.1127.
  10. Kaushik Chakrabarti and Sharad Mehrotra. Local dimensionality reduction: A new approach to indexing high dimensional spaces. In VLDB 2000, Proceedings of 26th International Conference on Very Large Data Bases, September 10-14, 2000, Cairo, Egypt, pages 89-100. Morgan Kaufmann, 2000. URL: http://www.vldb.org/conf/2000/P089.pdf.
  11. Timothy M. Chan. Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom., 35(1-2):20-35, 2006. URL: https://doi.org/10.1016/J.COMGEO.2005.10.002.
  12. Chaeyoon Chung, Taehoon Ahn, Sang Won Bae, and Hee-Kap Ahn. Parallel line centers with guaranteed separation. Comput. Geom., 129:102185, 2025. URL: https://doi.org/10.1016/J.COMGEO.2025.102185.
  13. Vasek Chvátal. A greedy heuristic for the set-covering problem. Math. Oper. Res., 4(3):233-235, 1979. URL: https://doi.org/10.1287/MOOR.4.3.233.
  14. Greg N. Frederickson and Donald B. Johnson. The complexity of selection and ranking in X+Y and matrices with sorted columns. J. Comput. Syst. Sci., 24(2):197-208, 1982. URL: https://doi.org/10.1016/0022-0000(82)90048-4.
  15. Greg N. Frederickson and Donald B. Johnson. Generalized selection and ranking: Sorted matrices. SIAM J. Comput., 13(1):14-30, 1984. URL: https://doi.org/10.1137/0213002.
  16. Sariel Har-Peled and Kasturi R. Varadarajan. Projective clustering in high dimensions using core-sets. In Proceedings of the 18th Annual Symposium on Computational Geometry, Barcelona, Spain, June 5-7, 2002, pages 312-318. ACM, 2002. URL: https://doi.org/10.1145/513400.513440.
  17. Refael Hassin and Nimrod Megiddo. Approximation algorithms for hitting objects with straight lines. Discret. Appl. Math., 30(1):29-42, 1991. URL: https://doi.org/10.1016/0166-218X(91)90011-K.
  18. Piotr Indyk, Rajeev Motwani, Prabhakar Raghavan, and Santosh S. Vempala. Locality-preserving hashing in multidimensional spaces. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 618-625. ACM, 1997. URL: https://doi.org/10.1145/258533.258656.
  19. Jerzy W. Jaromczyk and Miroslaw Kowaluk. The two-line center problem from a polar view: A new algorithm and data structure. In Algorithms and Data Structures, 4th International Workshop, WADS '95, Kingston, Ontario, Canada, August 16-18, 1995, Proceedings, volume 955 of Lecture Notes in Computer Science, pages 13-25. Springer, 1995. URL: https://doi.org/10.1007/3-540-60220-8_47.
  20. D. T. Lee and Ying-Fung Wu. Geometric complexity of some loction problems. Algorithmica, 1(2):193-211, 1986. URL: https://doi.org/10.1007/BF01840442.
  21. Nimrod Megiddo and Arie Tamir. On the complexity of locating linear facilities in the plane. Oper. Res. Lett., 1(5):194-197, 1982. URL: https://doi.org/10.1016/0167-6377(82)90039-6.
  22. Franco P. Preparata and Michael Ian Shamos. Computational Geometry - An Introduction. Texts and Monographs in Computer Science. Springer, New Haven, CT, USA, 1985. UMI Order No. 78-18546. URL: https://doi.org/10.1007/978-1-4612-1098-6.
  23. Cecilia M. Procopiuc. Projective clustering. In Encyclopedia of Machine Learning and Data Mining, pages 1018-1025. Springer, 2017. URL: https://doi.org/10.1007/978-1-4899-7687-1_676.
  24. Cecilia Magdalena Procopiuc, Michael Jones, Pankaj K. Agarwal, and T. M. Murali. A monte carlo algorithm for fast projective clustering. In Proceedings of the 2002 ACM SIGMOD International Conference on Management of Data, Madison, Wisconsin, USA, June 3-6, 2002, pages 418-427. ACM, 2002. URL: https://doi.org/10.1145/564691.564739.
  25. Haitao Wang. Dynamic convex hulls under window-sliding updates. Comput. Geom. Topol., 4(1):3:1-3:14, 2025. URL: https://doi.org/10.57717/CGT.V4I1.53.
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
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact