Discrete & Computational Geometry

, Volume 56, Issue 4, pp 836–859 | Cite as

A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon

  • Hee-Kap Ahn
  • Luis Barba
  • Prosenjit Bose
  • Jean-Lou De Carufel
  • Matias Korman
  • Eunjin Oh
Article
  • 194 Downloads

Abstract

Let P be a closed simple polygon with n vertices. For any two points in P, the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P. The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P. In 1989, Pollack et al. (Discrete Comput Geom 4(1): 611–626, 1989) showed an \(O(n\log n)\)-time algorithm that computes the geodesic center of P. Since then, a longstanding question has been whether this running time can be improved. In this paper we affirmatively answer this question and present a deterministic linear-time algorithm to solve this problem.

Keywords

Geodesic distance Facility location Simple polygons 

Mathematics Subject Classification

65D18 

References

  1. 1.
    Aronov, B.: On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica 4(1–4), 109–140 (1989)MathSciNetCrossRefMATHGoogle 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.: Computing the geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University (1985)Google Scholar
  4. 4.
    Bae, S.W., Korman, M., Mitchell, J.S.B., Okamoto, Y., Polishchuk, V., Wang, H.: Computing the L1 geodesic diameter and center of a polygonal domain. In: Ollinger, N., Vollmer, H. (eds) 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 47, pp. 14:1–14:14. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Dagstuhl (2016). ISBN 978-3-95977-001-9Google Scholar
  5. 5.
    Bae, S.W., Korman, M., Okamoto, Y.: The geodesic diameter of polygonal domains. Discrete Comput. Geom. 50(2), 306–329 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bae, S.W., Korman, M., Okamoto, Y.: Computing the geodesic centers of a polygonal domain. In: Computational Geometry: Theory and Applications. Special issue of selected papers from the 26th Canadian Conference on Computational Geometry (CCCG’14) (2015) (in press)Google Scholar
  7. 7.
    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)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chazelle B.: A theorem on polygon cutting with applications. In: Proceedings of FOCS, pp. 339–349 (1982)Google Scholar
  9. 9.
    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6(1), 485–524 (1991)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Djidjev, H., Lingas, A., Sack, J.-R.: An \(O(n\log n)\) algorithm for computing the link center of a simple polygon. Discrete Comput. Geom. 8, 131–152 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Trans. Graph. 9(1), 66–104 (1990)CrossRefMATHGoogle Scholar
  12. 12.
    Guibas, L., Hershberger, J., Leven, D., Sharir, M., Tarjan, R.E.: Linear-time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica 2(1–4), 209–233 (1987)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Harborth, H., Möller, M.: The Esther-Klein-problem in the projective plane. Inst. Mathematik, TU Braunschweig (1993)Google Scholar
  14. 14.
    Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hershberger, J., Suri, S.: Matrix searching with the shortest-path metric. SIAM J. Comput. 26(6), 1612–1634 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ke, Y.: An efficient algorithm for link-distance problems. In: Proceedings of the Fifth Annual Symposium on Computational Geometry, pp. 69–78. ACM, Saarbrücken (1989)Google Scholar
  17. 17.
    Lee, D.-T., Preparata, F.P.: Euclidean shortest paths in the presence of rectilinear barriers. Networks 14(3), 393–410 (1984)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Matoušek, J.: Approximations and optimal geometric divide-and-conquer. In: Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, pp. 505–511. ACM, New York (1991)Google Scholar
  19. 19.
    Matoušek, J.: Approximations and optimal geometric divide-and-conquer. J. Comput. Syst. Sci. 50(2), 203–208 (1995)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Matoušek, J.: Lectures on Discrete Geometry, Graduate Texts in Math, vol. 212. Springer, New York (2002)Google Scholar
  21. 21.
    Megiddo, N.: On the ball spanned by balls. Discrete Comput. Geom. 4(1), 605–610 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier, Amsterdam (2000)CrossRefGoogle Scholar
  23. 23.
    Nilsoon, B.J., Schuierer, S.: Computing the rectilinear link diameter of a polygon. In: Computational Geometry—Methods, Algorithms and Applications, pp. 203–215. Springer, Berlin (1991)Google Scholar
  24. 24.
    Nilsoon, B.J., Schuierer, S.: An optimal algorithm for the rectilinear link center of a rectilinear polygon. Comput. Geom. 6, 169–194 (1996)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(1), 611–626 (1989)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Suri, S.: Minimum link paths in polygons and related problems. PhD Thesis, Johns Hopkins University (1987)Google Scholar
  27. 27.
    Suri, S.: Computing geodesic furthest neighbors in simple polygons. J. Comput. Syst. Sci. 39(2), 220–235 (1989)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Turán, P.: On an extremal problem in graph theory. Matematikai és Fizikai Lapok 48(436–452), 137 (1941)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Luis Barba
    • 2
    • 3
  • Prosenjit Bose
    • 2
  • Jean-Lou De Carufel
    • 2
  • Matias Korman
    • 4
  • Eunjin Oh
    • 1
  1. 1.Department of Computer Science and EngineeringPOSTECHPohangKorea
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Tohoku UniversitySendaiJapan

Personalised recommendations