Advertisement

Algorithmica

pp 1–40 | Cite as

The Geodesic Farthest-Point Voronoi Diagram in a Simple Polygon

  • Eunjin Oh
  • Luis Barba
  • Hee-Kap AhnEmail author
Article
First Online:
  • 13 Downloads

Abstract

Given a set of point sites in a simple polygon, the geodesic farthest-point Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an \(O(n\log \log n+ m\log m)\)-time algorithm to compute the geodesic farthest-point Voronoi diagram of m point sites in a simple n-gon. This improves the previously best known algorithm by Aronov et al. (Discrete Comput Geom 9(3):217–255, 1993). In the case that all point sites are on the boundary of the simple polygon, we can compute the geodesic farthest-point Voronoi diagram in \(O((n+m) \log \log n)\) time.

Keywords

Farthest-point Voronoi diagram Simple polygon Geodesic metric 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

Funding was provided by Ministry of Science and ICT (KR) (Grant No. IITP-2017-0-00905).

References

  1. 1.
    Aggarwal, A., Guibas, L.J., Saxe, J., Shor, P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete Comput. Geom. 4(6), 591–604 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ahn, H.-K., Barba, L., Bose, P., 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
  3. 3.
    Aronov, B., Fortune, S., Wilfong, G.: The furthest-site geodesic Voronoi diagram. Discrete Comput. Geom. 9(3), 217–255 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Asano, T., Toussaint, G.: Computing the geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University (1985)Google Scholar
  5. 5.
    Chazelle, B.: A theorem on polygon cutting with applications. In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science (FOCS 1982), pp. 339–349 (1982)Google Scholar
  6. 6.
    Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Hershberger, J., Sharir, M., Snoeyink, J.: Ray shooting in polygons using geodesic triangulations. Algorithmica 12(1), 54–68 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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), 209–233 (1987)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Guibas, L.J., Hershberger, J.: Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci. 39(2), 126–152 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hershberger, J., Suri, S.: Matrix searching with the shortest-path metric. SIAM J. Comput. 26(6), 1612–1634 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Klein, R.: Concrete and Abstract Voronoi Diagrams. Springer, Berlin (1989)CrossRefGoogle Scholar
  11. 11.
    Klein, R., Lingas, A.: Hamiltonian abstract Voronoi diagrams in linear time. In: Prooceedings of the 5th International Symposium on Algorithms and Computation (ISAAC 1994), pp. 11–19. Springer, Berlin (1994)CrossRefGoogle Scholar
  12. 12.
    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Handbook of Computational Geometry, pp. 633–701. Elsevier (2000)Google Scholar
  13. 13.
    Oh, E., Ahn, H.-K.: Voronoi diagrams for a moderate-sized point-set in a simple polygon. In: Proceedings of the 33rd International Symposium on Computational Geometry (SoCG 2017), vol. 77, pp. 52:1–52:15. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2017)Google Scholar
  14. 14.
    Oh, E., Barba, L., Ahn, H.-K.: The farthest-point geodesic Voronoi diagram of points on the boundary of a simple polygon. In: Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016), vol. 51, pp. 56:1–56:15. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2016)Google Scholar
  15. 15.
    Papadopoulou, E.: \(k\)-pairs non-crossing shortest paths in a simple polygon. Int. J. Comput. Geom. Appl. 9(6), 533–552 (1999)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(6), 611–626 (1989)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Suri, S.: Computing geodesic furthest neighbors in simple polygons. J. Comput. Syst. Sci. 39(2), 220–235 (1989)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Toussaint, G.T.: An optimal algorithm for computing the relative convex hull of a set of points in a polygon. In: Proceeding of EURASIP-86, Part 2, pp. 853–856 (1986)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Pohang University of Science and TechnologyPohangKorea
  2. 2.Department of Computer ScienceETH ZürichZurichSwitzerland

Personalised recommendations