Discrete & Computational Geometry

, Volume 62, Issue 4, pp 743–774 | Cite as

\(L_1\) Geodesic Farthest Neighbors in a Simple Polygon and Related Problems

  • Sang Won BaeEmail author


We investigate the \(L_1\) geodesic farthest neighbors in a simple polygon P, and address several fundamental problems related to farthest neighbors. Given a subset \(S \subseteq P\), an \(L_1\) geodesic farthest neighbor of \(p \in P\) from S is one that maximizes the length of \(L_1\) shortest path from p in P. Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the \(L_1\) geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset \(S \subseteq P\) with respect to the \(L_1\) geodesic distance after removing redundancy.


\(L_1\) geodesic distance Simple polygon Farthest neighbor Diameter Radius Center Two-center 

Mathematics Subject Classification

68U05 68Q25 68W05 



  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. Discrete Comput. Geom. 56(4), 836–859 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aronov, B., Fortune, S., Wilfong, G.: The furthest-site geodesic Voronoĭ diagram. Discrete Comput. Geom. 9(3), 217–255 (1993)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bae, S.W., Korman, M., Okamoto, Y., Wang, H.: Computing the $L_1$ geodesic diameter and center of a simple polygon in linear time. Comput. Geom. 48(6), 495–505 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barba, L.: Optimal algorithm for geodesic farthest-point Voronoi diagrams. In: Proceedings of the 35th Symposium on Computational Geometry (SoCG 2019). LIPIcs (2019)Google Scholar
  5. 5.
    Chan, T.M.: More planar two-center algorithms. Comput. Geom. 13(3), 189–198 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Drezner, Z.: On the rectangular $p$-center problem. Nav. Res. Logist. 34(2), 229–234 (1987)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guibas, L.J., Hershberger, J.: Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci. 39(2), 126–152 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guibas, L.J., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.: Linear time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2(2), 209–233 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. 4(2), 63–97 (1994)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hershberger, J., Suri, S.: Matrix searching with the shortest path metric. SIAM J. Comput. 26(6), 1612–1634 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Megiddo, N.: Linear-time algorithms for linear programming in ${ R}^3$ and related problems. SIAM J. Comput. 12(4), 759–776 (1983)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Oh, E., Barba, L., Ahn, H.K.: The farthest-point geodesic Voronoi diagram of points on the boundary of a simple polygon. In: Fekete, S., Lubiw, A. (eds.). Proceedings of the 32nd Symposium on Computational Geometry (SoCG 2016), vol. 51, pp. 56:1–56:15. LIPIcs, Leibniz International Proceedings in Informatics, Schloss Dagstuhl. Leibniz-Zentrum für Informatik, Wadern (2016)Google Scholar
  13. 13.
    Oh, E., Bae, S.W., Ahn, H.-K.: Computing a geodesic two-center of points in a simple polygon. Comput. Geom. 82, 45–59 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Oh, E., De Carufel, J.L., Ahn, H.K.: The 2-center problem in a simple polygon. Comput. Geom. 74, 21–37 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Texts and Monographs in Computer Science. Springer, New York (1985)CrossRefGoogle Scholar
  16. 16.
    Toussaint, G.T.: Computing geodesic properties inside a simple polygon. Revue D’Intelligence Artificielle 3, 9–42 (1989)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Division of Computer Science and EngineeringKyonggi UniversitySuwonKorea

Personalised recommendations