Document Open Access Logo

Optimal Algorithm for the Planar Two-Center Problem

Authors Kyungjin Cho , Eunjin Oh , Haitao Wang , Jie Xue

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.40.pdf
  • Filesize: 1 MB
  • 15 pages

Document Identifiers

Author Details

Kyungjin Cho
  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
Eunjin Oh
  • Department of Computer Science and Engineering, POSTECH, Pohang, South Korea
Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Jie Xue
  • Department of Computer Science, New York University Shanghai, China

Funding

  • Cho, Kyungjin: Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2023-00209069).
  • Oh, Eunjin: Supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.RS-2023-00209069).
  • Wang, Haitao: Supported in part by NSF under Grant CCF-2300356.

Cite As Get BibTex

Kyungjin Cho, Eunjin Oh, Haitao Wang, and Jie Xue. Optimal Algorithm for the Planar Two-Center Problem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.40

Abstract

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set S of n points in the plane and the goal is to find two smallest congruent disks whose union contains all points of S. A longstanding open problem has been to obtain an O(nlog n)-time algorithm for planar two-center, matching the Ω(nlog n) lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in O(nlog² n) time. In this paper, we present an O(nlog n)-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • two-center
  • r-coverage
  • disk coverage
  • circular hulls

Metrics

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

Related Versions

References

  1. Pankaj K. Agarwal and Jeff M. Phillips. An efficient algorithm for 2D Euclidean 2-center with outliers. In Proceedings of the 16th Annual European Symposium on Algorithms (ESA), pages 64-75. Springer, 2008. Google Scholar
  2. Pankaj K. Agarwal and Micha Sharir. Planar geometric location problems. Algorithmica, 11(2):185-195, 1994. Google Scholar
  3. Pankaj K. Agarwal, Micha Sharir, and Emo Welzl. The discrete 2-center problem. Discrete and Computational Geometry., 20(3):287-305, 1998. Google Scholar
  4. Esther M. Arkin, José Miguel Díaz-Báñez, Ferran Hurtado, Piyush Kumar, Joseph S.B. Mitchell, Belén Palop, Pablo Pérez-Lantero, Maria Saumell, and Rodrigo I. Silveira. Bichromatic 2-center of pairs of points. Computational Geometry: Theory and Applications, 48(2):94-107, 2015. Google Scholar
  5. Timothy M. Chan. More planar two-center algorithms. Computational Geometry: Theory and Applications, 13(3):189-198, 1999. Google Scholar
  6. Bernard Chazelle and Jiří Matoušek. On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms, 21:579-597, 1996. Google Scholar
  7. Kyungjin Cho, Eunjin Oh, Haitao Wang, and Jie Xue. Optimal algorithm for the planar two-center problem, 2023. URL: https://arxiv.org/abs/2007.08784.
  8. Jongmin Choi and Hee-Kap Ahn. Efficient planar two-center algorithms. Computational Geometry: Theory and Applications, 97(101768), 2021. Google Scholar
  9. Jongmin Choi, Jaegun Lee, and Hee-Kap Ahn. Efficient k-center algorithms for planar points in convex position. In Proceedings of the 18th Algorithms and Data Structures Symposium (WADS), pages 262-274, 2023. Google Scholar
  10. Richard Cole. Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34(1):200-208, 1987. Google Scholar
  11. Mark de Berg, Marcel Roeloffzen, and Bettina Speckmann. Kinetic 2-centers in the black-box model. In Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry (SoCG), pages 145-154, 2013. Google Scholar
  12. M.E. Dyer. On a multidimensional search technique and its application to the Euclidean one centre problem. SIAM Journal on Computing, 15(3):725-738, 1986. Google Scholar
  13. Herbert Edelsbrunner, David G. Kirkpatrick, and Raimund Seidel. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29:551-559, 1983. Google Scholar
  14. David Eppstein. Dynamic three-dimensional linear programming. In Proceedings of the 32nd Annual Symposium of Foundations of Computer Science (FOCS), pages 488-494, 1991. Google Scholar
  15. David Eppstein. Faster construction of planar two-centers. In Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 131-138, 1997. Google Scholar
  16. Dan Halperin, Micha Sharir, and Ken Goldberg. The 2-center problem with obstacles. Journal of Algorithms, 42(1):109-134, 2002. Google Scholar
  17. John Hershberger. A faster algorithm for the two-center decision problem. Information Processing Letters, 1:23-29, 1993. Google Scholar
  18. John Hershberger and Subhash Suri. Finding tailored partitions. Journal of Algorithms, 12(3):431-463, 1991. Google Scholar
  19. R. Z. Hwang, R. C. T. Lee, and R. C. Chang. The slab dividing approach to solve the Euclidean p-center problem. Algorithmica, 9(1):1-22, 1993. Google Scholar
  20. Jerzy W. Jaromczyk and Mirosław Kowaluk. An efficient algorithm for the Euclidean two-center problem. In Proceedings of the Tenth Annual Symposium on Computational Geometry (SoCG), pages 303-311. Association for Computing Machinery, 1994. Google Scholar
  21. Matthew J. Katz and Micha Sharir. An expander-based approach to geometric optimization. SIAM Journal on Computing, 26(5):1384-1408, 1997. Google Scholar
  22. Nimrod Megiddo. Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM, 30(4):852-865, 1983. Google Scholar
  23. Nimrod Megiddo. Linear-time algorithms for linear programming in R³ and related problems. SIAM Journal on Computing, 12(4):759-776, 1983. Google Scholar
  24. Nimrod Megiddo. Linear programming in linear time when the dimension is fixed. Journal of the ACM, 31(1):114-127, 1984. Google Scholar
  25. Eunjin Oh, Sang Won Bae, and Hee-Kap Ahn. Computing a geodesic two-center of points in a simple polygon. Computational Geomtry: Theory and Computation, 82:45-59, 2019. Google Scholar
  26. Eunjin Oh, Jean-Lou De Carufel, and Hee-Kap Ahn. The geodesic 2-center problem in a simple polygon. Computational Geomtry: Theory and Computation, 74:21-37, 2018. Google Scholar
  27. Micha Sharir. A near-linear algorithm for the planar 2-center problem. In Proceedings of the Twelfth Annual Symposium on Computational Geometry (SoCG), pages 106-112, 1996. Google Scholar
  28. Xuehou Tan and Bo Jiang. Simple O(nlog²n) algorithms for the planar 2-center problem. In Proceedings of the 23rd International Computing and Combinatorics Conference (COCOON), pages 481-491, 2017. Google Scholar
  29. Haitao Wang. On the planar two-center problem and circular hulls. Discrete and Computational Geometry, 68(4):1175-1226, 2022. Google Scholar
  30. Haitao Wang. Unit-disk range searching and applications. In Proceedings of the 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 32:1-32:17, 2022. Google Scholar
  31. Haitao Wang and Jie Xue. Improved algorithms for the bichromatic two-center problem for pairs of points. Computational Geometry: Theory and Applications, 100(101806):1-12, 2022. Google Scholar
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact