Discrete & Computational Geometry

, Volume 57, Issue 3, pp 674–701 | Cite as

Computing the \(L_1\) Geodesic Diameter and Center of a Polygonal Domain

  • Sang Won Bae
  • Matias Korman
  • Joseph S. B. Mitchell
  • Yoshio Okamoto
  • Valentin Polishchuk
  • Haitao Wang
Article

Abstract

For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the \(L_1\) geodesic diameter in \(O(n^2+h^4)\) time and the \(L_1\) geodesic center in \(O((n^4+n^2 h^4)\alpha (n))\) time, respectively, where \(\alpha (\cdot )\) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in \(O(n^{7.73})\) or \(O(n^7(h+\log n))\) time, and compute the geodesic center in \(O(n^{11}\log n)\) time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on \(L_1\) shortest paths in polygonal domains.

Keywords

Geodesic diameter Geodesic center Shortest paths Polygonal domains \(L_1\) metric 

Mathematics Subject Classification

68W05 68W40 

References

  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 on Computational Geometry, pp. 209–223 (2015)Google Scholar
  2. 2.
    Aronov, B.: On the geodesic Voronoi diagram of point sites in a simple polygon. Algorithmica 4(1–4), 109–140 (1989)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Asano, T., Toussaint, G.: Computing the geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University, Montreal (1985)Google Scholar
  4. 4.
    Bae, S.W., Korman, M., Okamoto, Y.: The geodesic diameter of polygonal domains. Discrete Comput. Geom. 50, 306–329 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bae, S.W., Korman, M., Okamoto, Y.: Computing the geodesic centers of a polygonal domain. In: Proceedings of the 26th Canadian Conference on Computational Geometry (2014). Journal version to appear in Computational Geometry: Theory and Applications. doi:10.1016/j.comgeo.2015.10.009
  6. 6.
    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, 495–505 (2015)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bar-Yehuda, R., Chazelle, B.: Triangulating disjoint Jordan chains. Int. J. Comput. Geom. Appl. 4(4), 475–481 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chazelle, B.: A theorem on polygon cutting with applications. In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, pp. 339–349 (1982)Google Scholar
  9. 9.
    Chen, D.Z., Wang, H.: A nearly optimal algorithm for finding \(L_1\) shortest paths among polygonal obstacles in the plane. In: Proceedings of the 19th European Symposium on Algorithms, pp. 481–492 (2011)Google Scholar
  10. 10.
    Chen, D.Z., Wang, H.: Computing the visibility polygon of an island in a polygonal domain. In: Proceedings of the 39th International Colloquium on Automata, Languages and Programming, pp. 218–229 (2012). Journal version published online in Algorithmica, 2015Google Scholar
  11. 11.
    Chen, D.Z., Wang, H.: \(L_1\) shortest path queries among polygonal obstacles in the plane. In: Proceedings of the 30th Symposium on Theoretical Aspects of Computer Science, pp. 293–304 (2013)Google Scholar
  12. 12.
    Edelsbrunner, H., Guibas, L.J., Sharir, M.: The upper envelope of piecewise linear functions: algorithms and applications. Discrete Comput. Geom. 4, 311–336 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Guibas, L.J., 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
  14. 14.
    Hershberger, J., Snoeyink, J.: Computing minimum length paths of a given homotopy class. Comput. Geom. 4(2), 63–97 (1994)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.
    Inkulu, R., Kapoor, S.: Planar rectilinear shortest path computation using corridors. Comput. Geom. 42(9), 873–884 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kapoor, S., Maheshwari, S.N., Mitchell, J.S.B.: An efficient algorithm for Euclidean shortest paths among polygonal obstacles in the plane. Discrete Comput. Geom. 18(4), 377–383 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mitchell, J.S.B.: An optimal algorithm for shortest rectilinear paths among obstacles. In: The 1st Canadian Conference on Computational Geometry (1989)Google Scholar
  19. 19.
    Mitchell, J.S.B.: \(L_1\) shortest paths among polygonal obstacles in the plane. Algorithmica 8(1), 55–88 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Papadopoulou, E., Lee, D.T.: A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica 20(4), 319–352 (1998)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(1), 611–626 (1989)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Schuierer, S.: Computing the \(L_1\)-diameter and center of a simple rectilinear polygon. In: Proceedings of the International Conference on Computing and Information, pp. 214–229 (1994)Google Scholar
  23. 23.
    Suri, S.: Computing geodesic furthest neighbors in simple polygons. J. Comput. Syst. Sci. 39, 220–235 (1989)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wang, H.: On the geodesic centers of polygonal domains. In: Proceedings of the 24th European Symposium on Algorithms, pp. 77:1–77:17 (2016)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Sang Won Bae
    • 1
  • Matias Korman
    • 2
  • Joseph S. B. Mitchell
    • 3
  • Yoshio Okamoto
    • 4
  • Valentin Polishchuk
    • 5
  • Haitao Wang
    • 6
  1. 1.Kyonggi UniversitySuwonSouth Korea
  2. 2.Tohoku UniversitySendaiJapan
  3. 3.Stony Brook UniversityNew YorkUSA
  4. 4.The University of Electro-CommunicationsTokyoJapan
  5. 5.Linköping UniversityLinköpingSweden
  6. 6.Utah State UniversityLoganUSA

Personalised recommendations