Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
167 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Optimal Algorithm for the Planar Two-Center Problem (2007.08784v5)

Published 17 Jul 2020 in cs.CG

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 $O(n\log n)$-time algorithm for planar two-center, matching the $\Omega(n\log n)$ 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 $O(n\log2 n)$ time. In this paper, we present an $O(n\log n)$-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (31)
  1. An efficient algorithm for 2D Euclidean 2-center with outliers. In Proceedings of the 16th Annual European Symposium on Algorithms (ESA), pages 64–75, 2008.
  2. Planar geometric location problems. Algorithmica, 11(2):185–195, 1994.
  3. The discrete 2-center problem. Discrete and Computational Geometry., 20(3):287–305, 1998.
  4. Bichromatic 2-center of pairs of points. Computational Geometry: Theory and Applications, 48(2):94–107, 2015.
  5. An efficient algorithm of the planar 3-center problem for a set of the convex position points. In MATEC Web of Conferences, volume 232, page 03022. EDP Sciences, 2018.
  6. Timothy M. Chan. More planar two-center algorithms. Computational Geometry: Theory and Applications, 13(3):189–198, 1999.
  7. On linear-time deterministic algorithms for optimization problems in fixed dimension. Journal of Algorithms, 21:579–597, 1996.
  8. Efficient planar two-center algorithms. Computational Geometry: Theory and Applications, 97(101768), 2021.
  9. Efficient k𝑘kitalic_k-center algorithms for planar points in convex position. In Proceedings of the 18th Algorithms and Data Structures Symposium (WADS), pages 262–274, 2023.
  10. Richard Cole. Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM, 34(1):200–208, 1987.
  11. Kinetic 2-centers in the black-box model. In Proceedings of the Twenty-Ninth Annual Symposium on Computational Geometry (SoCG), pages 145–154, 2013.
  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.
  13. On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29:551–559, 1983.
  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.
  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.
  16. The 2-center problem with obstacles. Journal of Algorithms, 42(1):109–134, 2002.
  17. John Hershberger. A faster algorithm for the two-center decision problem. Information Processing Letters, 1:23–29, 1993.
  18. Finding tailored partitions. Journal of Algorithms, 12(3):431–463, 1991.
  19. The slab dividing approach to solve the Euclidean p𝑝pitalic_p-center problem. Algorithmica, 9(1):1–22, 1993.
  20. An efficient algorithm for the Euclidean two-center problem. In Proceedings of the Tenth Annual Symposium on Computational Geometry (SoCG), page 303–311, 1994.
  21. An expander-based approach to geometric optimization. SIAM Journal on Computing, 26(5):1384–1408, 1997.
  22. Nimrod Megiddo. Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM, 30(4):852––865, 1983.
  23. Nimrod Megiddo. Linear-time algorithms for linear programming in R3superscript𝑅3R^{3}italic_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and related problems. SIAM Journal on Computing, 12(4):759–776, 1983.
  24. Nimrod Megiddo. Linear programming in linear time when the dimension is fixed. Journal of the ACM, 31(1):114–127, 1984.
  25. Computing a geodesic two-center of points in a simple polygon. Computational Geomtry: Theory and Computation, 82:45–59, 2019.
  26. The geodesic 2-center problem in a simple polygon. Computational Geomtry: Theory and Computation, 74:21–37, 2018.
  27. Micha Sharir. A near-linear algorithm for the planar 2-center problem. In Proceedings of the Twelfth Annual Symposium on Computational Geometry (SoCG), page 106–112, 1996.
  28. Xuehou Tan and Bo Jiang. Simple O⁢(n⁢log2⁡n)𝑂𝑛superscript2𝑛O(n\log^{2}n)italic_O ( italic_n roman_log start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_n ) algorithms for the planar 2-center problem. In Proceedings of the 23rd International Computing and Combinatorics Conference (COCOON), pages 481–491, 2017.
  29. Haitao Wang. On the planar two-center problem and circular hulls. Discrete and Computational Geometry, 68(4):1175–1226, 2022.
  30. Haitao Wang. Unit-disk range searching and applications. Journal of Computational Geometry, 14:343–394, 2023.
  31. Improved algorithms for the bichromatic two-center problem for pairs of points. Computational Geometry: Theory and Applications, 100(101806):1–12, 2022.
Citations (3)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets