Discrete & Computational Geometry

, Volume 50, Issue 2, pp 306–329 | Cite as

The Geodesic Diameter of Polygonal Domains



This paper studies the geodesic diameter of polygonal domains having \(h\) holes and \(n\) corners. For simple polygons (i.e., \(h=0\)), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as shown by Hershberger and Suri. For general polygonal domains with \(h \ge 1\), however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time \(O(n^{7.73})\) or \(O(n^7 (\log n + h))\). The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.


Polygonal domain Shortest path Geodesic diameter Exact algorithm  Convex function Lower envelope 



We thank Hee-Kap Ahn, Jiongxin Jin, Christian Knauer, and Joseph Mitchell for fruitful discussion. We also thank Joseph O’Rourke for pointing out the reference [21]. Work by S.W. Bae was supported by National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No. 2010-0005974 and NRF-2013R1A1A1A05006927). M. Korman received the support of the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia, the European Union, and ESF EUROCORES Programme EuroGIGA-ComPoSe IP04-MICINN Project EUI-EURC-2011-4306. Work by Y. Okamoto was supported by Global COE Program “Computationism as a Foundation for the Sciences” and Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan, and Japan Society for the Promotion of Science.


  1. 1.
    Agarwal, P.K., Aronov, B., O’Rourke, J., Schevon, C.A.: Star unfolding of a polytope with applications. SIAM J. Comput. 26(6), 1689–1713 (1997)Google Scholar
  2. 2.
    Aronov, B., Fortune, S., Wilfong, G.: The furthest-site geodesic Voronoi diagram. Discrete Comput. Geom. 9, 217–255 (1993)MathSciNetMATHCrossRefGoogle 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., Chwa, K.-Y.: The geodesic farthest-site Voronoi diagram in a polygonal domain with holes. In: Proceedings of 25th Annual Symposium Computational Geometry (SoCG), pp. 198–207 (2009)Google Scholar
  5. 5.
    Bae, S.W., Okamoto, Y.: Querying two boundary points for shortest paths in a polygonal domain. Comput. Geom. 45(7), 284–293 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bae, S.W., Korman, M., Okamoto, Y.: The geodesic diameter of polygonal domains. In: Proceedings of 18th Annual European Symposium on Algorithms. Part 1. Lecture Notes in Computer Science, vol. 6346, pp. 500–511 (2010)Google Scholar
  7. 7.
    Chazelle, B.: A theorem on polygon cutting with applications. In: Proceedings of 23rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 339–349 (1982)Google Scholar
  8. 8.
    Chiang, Y.-J., Mitchell, J.S.B.: Two-point Euclidean shortest path queries in the plane. In: Proceedings of 10th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 215–224 (1999)Google Scholar
  9. 9.
    Cook, A.F., IV, Wenk, C.: Shortest path problems on a polyhedral surface. In: Proceedings of 11th International Symposium on Algorithms and Data Structures (WADS), pp. 156–167 (2009)Google Scholar
  10. 10.
    Guibas, L.J., Hershberger, J.: Optimal shortest path queries in a simple polygon. J. Comput. Syst. Sci. 39(2), 126–152 (1989)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Guo, H., Maheshwari, A., Sack, J.-R.: Shortest path queries in polygonal domains. In: Proceedings of the 4th International Conference on Algorithmic Aspects in Information and Management (AAIM). Lecture Notes in Computer Science, vol. 5034, pp. 200–211 (2008)Google Scholar
  12. 12.
    Hershberger, J., Suri, S.: Matrix searching with the shortest path metric. SIAM J. Comput. 26(6), 1612–1634 (1997)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28(6), 2215–2256 (1999)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Koivisto, M., Polishchuk, V.: Geodesic diameter of a polygonal domain in \(O(n^4\log n)\) time. CoRR, http://arxiv/abs/1006.1998 (2010)
  15. 15.
    Mitchell, J.S.B.: Shortest paths among obstacles in the plane. Int. J. Comput. Geom. Appl. 6(3), 309–331 (1996)MATHCrossRefGoogle Scholar
  16. 16.
    Mitchell, J.S.B.: Shortest paths and networks. In: Handbook of Discrete and Computational Geometry, 2nd edn, Chap. 27, pp. 607–641. CRC Press, Boca Raton (2004)Google Scholar
  17. 17.
    O’Rourke, J., Schevon, C.: Computing the geodesic diameter of a 3-polytope. In: Proceedings of the 5th Annual Symposium on Computational Geometry (SoCG), pp. 370–379 (1989)Google Scholar
  18. 18.
    O’Rourke, J., Suri, S.: Polygons. In: Handbook of Discrete and Computational Geometry, 2nd edn, Chap. 26, pp. 583–606. CRC Press, Boca Raton (2004)Google Scholar
  19. 19.
    Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(6), 611–626 (1989)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Suri, S.: The all-geodesic-furthest neighbors problem for simple polygons. In: Proceedings of 3rd Annual Symposium on Computational Geometry (SoCG), pp. 64–75 (1987)Google Scholar
  21. 21.
    Zalgaller, V.A.: An isoperimetric problem for tetrahedra. J. Math. Sci. 140(4), 511–527 (2007)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer ScienceKyonggi UniversitySuwonKorea
  2. 2.Department of Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaBarcelonaSpain
  3. 3.Department of Communication Engineering and InformaticsUniversity of Electro-CommunicationsChofuJapan

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