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Constrained Two-Line Center Problems

Authors Taehoon Ahn , Sang Won Bae

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • two-line center problem
  • geometric location problem
  • geometric optimization

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Abstract

Given a set P of n points in the plane, the two-line center problem asks to find two lines that minimize the maximum distance from each point in P to its closer one of the two resulting lines. The currently best algorithm for the problem takes O(n² log² n) time by Jaromczyk and Kowaluk in 1995. In this paper, we present faster algorithms for three variants of the two-line center problem in which the orientations of the resulting lines are constrained. Specifically, our algorithms solve the problem in O(n log n) time when the orientations of both lines are fixed; in O(n log³ n) time when the orientation of one line is fixed; and in O(n² α(n) log n) time when the angle between the two lines is fixed, where α(n) denotes the inverse Ackermann function.

Cite As Get BibTex

Taehoon Ahn and Sang Won Bae. Constrained Two-Line Center Problems. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.5

Author Details

Taehoon Ahn
  • Graduate School of Artificial Intelligence, Pohang University of Science and Technology, Republic of Korea
Sang Won Bae
  • Division of Artificial Intelligence and Computer Science, Kyonggi University, Suwon, Republic of Korea

Funding

  • Ahn, Taehoon: Supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) grant funded by the Korea government(MSIT) (No. 2019-0-01906).
  • Bae, Sang Won: Supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2023-00251168).

References

  1. Pankaj K. Agarwal, Cecilia M. Procopiuc, and Kasturi R. Varadarajan. A (1+ε)-approximation algorithm for 2-line-center. Computational Geometry: Theory and Applications, 26:119-128, 2003. URL: https://doi.org/10.1016/S0925-7721(03)00017-8.
  2. Pankaj K. Agarwal, Cecilia M. Procopiuc, and Kasturi R. Varadarajan. Approximation algorithms for a k-line center. Algorithmica, 42(3):221-230, 2005. URL: https://doi.org/10.1007/s00453-005-1166-x.
  3. Pankaj K. Agarwal and Micha Sharir. Off-line dynamic maintenance of the width of a planar point set. Computational Geometry: Theory and Applications, 1:65-78, 1991. URL: https://doi.org/10.1016/0925-7721(91)90001-U.
  4. Pankaj K. Agarwal and Micha Sharir. Planar geometric location problems and maintaining the width of a planar set. In Proceedings of the 2nd Annuual ACM-SIAM Symposium on Discrete Algorithms (SODA 1991), pages 449-458. SIAM, 1991. URL: http://dl.acm.org/citation.cfm?id=127787.127865.
  5. Pankaj K. Agarwal and Micha Sharir. Planar geometric location problems. Algorithmica, 11(2):185-195, 1994. URL: https://doi.org/10.1007/BF01182774.
  6. Pankaj K. Agarwal and Micha Sharir. Efficient randomized algorithms for some geometric optimization problems. Discrete & Computational Geometry, 16(4):317-337, 1996. URL: https://doi.org/10.1007/BF02712871.
  7. Taehoon Ahn and Sang Won Bae. Constrained two-line center problems, 2024. URL: https://arxiv.org/abs/2409.13304.
  8. Taehoon Ahn, Chaeyoon Chung, Hee-Kap Ahn, Sang Won Bae, Otfried Cheong, and Sang Duk Yoon. Minimum-width double-slabs and widest empty slabs in high dimensions. In J.A. Soto and A. Wiese, editors, Proceedings of the 16th Latin American Theoretical Informatics (LATIN 2024), Part I, volume 14578 of LNCS, pages 303-317, 2024. URL: https://doi.org/10.1007/978-3-031-55598-5_20.
  9. Takao Asano, Tetsuo Asano, Leonidas Guibas, John Hershberger, and Hiroshi Imai. Visibility of dijoint polygons. Algorithmica, 1:49-63, 1986. URL: https://doi.org/10.1007/BF01840436.
  10. Sang Won Bae. Minimum-width double-strip and parallelogram annulus. Theoretical Computer Science, 833:133-146, 2020. URL: https://doi.org/10.1016/j.tcs.2020.05.045.
  11. Gerth Stølting Brodal and Riko Jacob. Dynamic planar convex hull. In Proceedings of the 43rd Symposium on Foundations of Compututer Science (FOCS 2002), pages 617-626, 2002. URL: https://doi.org/10.1109/SFCS.2002.1181985.
  12. T. M. Chan. Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. International Journal of Computational Geometry and Applications, 12(1-2):67-85, 2002. URL: https://doi.org/10.1142/S0218195902000748.
  13. Thimothy M. Chan. A fully dynamic algorithm for planar width. Discrete & Computational Geometry, 30:17-24, 2003. URL: https://doi.org/10.1007/s00454-003-2923-8.
  14. Bernard Chazelle. Triangulating a simple polygon in linear time. Discrete & Computational Geometry, 6(3):485-524, 1991. URL: https://doi.org/10.1007/BF02574703.
  15. Chayoon Chung, Taehoon Ahn, Sang Won Bae, and Hee-Kap Ahn. Parallel line centers with guaranteed separation. In Proceedings of the 35th Canadian Conference on Computational Geometry (CCCG 2023), pages 153-160, 2023. Google Scholar
  16. Arun Kumar Das, Sandip Das, and Joydeep Mukherjee. Approximation algorithms for orthogonal line centers. Discrete Applied Mathematics, 338:69-76, 2023. URL: https://doi.org/10.1016/j.dam.2023.05.014.
  17. David Eppstein. Incremental and decremental maintenance of planar width. Journal of Algorithms, 37:570-577, 2000. URL: https://doi.org/10.1006/jagm.2000.1107.
  18. Alain Fournier and Delfin Y. Montuno. Triangulating simple polygons and equivalent problems. ACM Transactions on Graphics, 3:153-174, 1984. URL: https://doi.org/10.1145/357337.357341.
  19. Greg N. Frederickson and Donald B. Johnson. The complexity of selection and ranking in X+Y and matrices with sorted columns. Journal of Computer and System Science, 24:197-208, 1982. URL: https://doi.org/10.1016/0022-0000(82)90048-4.
  20. Greg N. Frederickson and Donald B. Johnson. Generalized selection and ranking: Sorted matrices. SIAM Journal on Computing, 13(1):14-30, 1984. URL: https://doi.org/10.1137/0213002.
  21. Alex Glozman, Klara Kedem, and Gregory Shpitalnik. On some geometric selection and optimization problems via sorted matrices. In Proceedings of the 4th International Workshop on Algorithms and Data Structures (WADS 1995), volume 955 of LNCS, pages 26-37, 1995. URL: https://doi.org/10.1007/3-540-60220-8_48.
  22. Alex Glozman, Klara Kedem, and Gregory Shpitalnik. On some geometric selection and optimization problems via sorted matrices. Computational Geometry: Theory and Applications, 11(1):17-28, 1998. URL: https://doi.org/10.1016/S0925-7721(98)00017-0.
  23. John Hershberger and Subhash Suri. Off-line maintenance of planar configurations. Journal of Algorithms, 21:453-475, 1991. URL: https://doi.org/10.1006/jagm.1996.0054.
  24. Jerzy Jaromczyk and Miroslaw Kowaluk. The two-line center problem from a polar view: A new algorithm and data structure. In Proceedings of the 4th International Workshop on Algorithms and Data Structures (WADS 1995), volume 955 of LNCS, pages 13-25, 1995. URL: https://doi.org/10.1007/3-540-60220-8_47.
  25. Matthew J. Katz, Klara Kedem, and Michael Segal. Constrained square-center problems. In Proceedings of the 6th Scandinavian Workshop on Algorithm Theory (SWAT 1998), volume 1432, pages 95-106. Springer, 1998. URL: https://doi.org/10.1007/BFb0054358.
  26. Matthew J. Katz and Micha Sharir. An expander-based approach to geometric optimization. SIAM Journal on Computing, 26(5):1384-1408, 1997. URL: https://doi.org/10.1137/S0097539794268649.
  27. Sang-Sub Kim, Sang Won Bae, and Hee-Kap Ahn. Covering a point set by two disjoint rectangles. International Journal of Computational Geometry & Applications, 21(3):313-330, 2011. URL: https://doi.org/10.1142/S0218195911003676.
  28. Nimrod Megiddo and Arie Tamir. On the complexity of locating linear facilities in the plane. Operations Research Letters, 1(5):194-197, 1982. URL: https://doi.org/10.1016/0167-6377(82)90039-6.
  29. Franco P. Preparata and Michael Ian Shamos. Computational Geometry: An Introduction. Springer Verlag, 1985. URL: https://doi.org/10.1007/978-1-4612-1098-6.
  30. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York, 1995. Google Scholar
  31. Godfried T. Toussaint. Solving geometric problems with the rotating calipers. In Proceedings of the 2nd Mediterranean Electrotechnical Conference (IEEE MELECON 1983), 1983. Google Scholar
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