Computing a Geodesic Two-Center of Points in a Simple Polygon

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


Given a simple polygon P and a set Q of points contained in P, we consider the geodesic k-center problem in which we seek to find k points, called centers, in P to minimize the maximum geodesic distance of any point of Q to its closest center. In this paper, we focus on the case for \(k=2\) and present the first exact algorithm that efficiently computes an optimal 2-center of Q with respect to the geodesic distance in P.


  1. 1.
    Ahn, H.K., Barba, L., Bose, P., De Carufel, J.L., Korman, M., Oh, E.: A linear-time algorithm for the geodesic center of a simple polygon. In: Proceedings of the 31st Symposium Computational Geometry (SoCG), vol. 34, pp. 209–223 (2015)Google Scholar
  2. 2.
    Aronov, B., Fortune, S., Wilfong, G.: The furthest-site geodesic voronoi diagram. Discrete Comput. Geom. 9(1), 217–255 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Asano, T., Toussaint, G.T.: Computing geodesic center of a simple polygon. Technical report SOCS-85.32, McGill University (1985)Google Scholar
  4. 4.
    Chan, T.M.: More planar two-center algorithms. Comput. Geom. 13(3), 189–198 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms 21(3), 579–597 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cole, R.: Parallel merge sort. SIAM J. Comput. 17(4), 770–785 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dyer, M.E.: On a multidimensional search technique and its application to the euclidean one-centre problem. SIAM J. Comput. 15(3), 725–738 (1986)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Feder, T., Greene, D.H.: Optimal algorithms for approximate clustering. In: Proceedings of the 20th ACM Symposium Theory Computing (STOC), pp. 434–444 (1988)Google Scholar
  9. 9.
    Guibas, L., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2(1–4), 209–233 (1987)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Halperin, D., Sharir, M., Goldberg, K.Y.: The 2-center problem with obstacles. J. Algorithms 42(1), 109–134 (2002)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hwang, R., Lee, R., Chang, R.: The slab dividing approach to solve the Euclidean \(p\)-center problem. Algorithmica 9(1), 1–22 (1993)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Megiddo, N.: Linear-time algorithms for linear programming in \(R^3\) and related problems. SIAM J. Comput. 12(4), 759–776 (1983)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Oh, E., De Carufel, J.-L., Ahn, H.-K.: The 2-center problem in a simple polygon. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 307–317. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48971-0_27 CrossRefGoogle Scholar
  14. 14.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(1), 611–626 (1989)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Toussaint, G.T.: Computing geodesic properties inside a simple polygon. Revue D’Intelligence Artificielle 3, 9–42 (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringPOSTECHPohangSouth Korea
  2. 2.Department of Computer ScienceKyonggi UniversitySuwonSouth Korea

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