ESA 2010: Algorithms – ESA 2010 pp 500-511

# The Geodesic Diameter of Polygonal Domains

• Sang Won Bae
• Matias Korman
• Yoshio Okamoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6346)

## Abstract

This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), it is known that the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time. For general polygonal domains with h ≥ 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithm that computes the geodesic diameter of a given polygonal domain in worst-case time O(n7.73) or O(n7 (logn + h)). Among other results, we show the following geometric observation: the geodesic diameter can be determined by two points in its interior. In such a case, there are at least five shortest paths between the points.

## Preview

### References

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)
2. 2.
Aronov, B., Fortune, S., Wilfong, G.: The furthest-site geodesic Voronoi diagram. Discrete Comput. Geom. 9, 217–255 (1993)
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., Chwa, K.-Y.: The geodesic farthest-site Voronoi diagram in a polygonal domain with holes. In: Proc. 25th Annu. Sympos. Comput. Geom. (SoCG), pp. 198–207 (2009)Google Scholar
5. 5.
Bae, S.W., Korman, M., Okamoto, Y.: The geodesic diameter of polygonal domains. arXiv preprint (2010), arXiv:1001.0695Google Scholar
6. 6.
Bae, S.W., Okamoto, Y.: Querying two boundary points for shortest paths in a polygonal domain. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 1054–1063. Springer, Heidelberg (2009), arXiv:0911.5017Google Scholar
7. 7.
Chazelle, B.: A theorem on polygon cutting with applications. In: Proc. 23rd Annu. Sympos. Found. Comput. Sci. (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: Proc. 10th ACM-SIAM Sympos. Discrete Algorithms (SODA), pp. 215–224 (1999)Google Scholar
9. 9.
Cook IV, A.F., Wenk, C.: Shortest path problems on a polyhedral surface. In: Proc. 11th Internat. Sympos. Algo. Data Struct (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)
11. 11.
Guo, H., Maheshwari, A., Sack, J.-R.: Shortest path queries in polygonal domains. In: Fleischer, R., Xu, J. (eds.) AAIM 2008. LNCS, vol. 5034, pp. 200–211. Springer, Heidelberg (2008)
12. 12.
Hershberger, J., Suri, S.: Matrix searching with the shortest path metric. SIAM J. Comput. 26(6), 1612–1634 (1997)
13. 13.
Hershberger, J., Suri, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28(6), 2215–2256 (1999)
14. 14.
Mitchell, J.S.B.: Shortest paths among obstacles in the plane. Internat. J. Comput. Geom. Appl. 6(3), 309–331 (1996)
15. 15.
Mitchell, J.S.B.: Shortest paths and networks. In: Handbook of Discrete and Computational Geometry, 2nd edn., ch. 27, pp. 607–641. CRC Press, Inc., Boca Raton (2004)Google Scholar
16. 16.
O’Rourke, J., Schevon, C.: Computing the geodesic diameter of a 3-polytope. In: Proc. 5th Annu. Sympos. Comput. Geom. (SoCG), pp. 370–379 (1989)Google Scholar
17. 17.
O’Rourke, J., Suri, S.: Polygons. In: Handbook of Discrete and Computational Geometry, 2nd edn., ch. 26, pp. 583–606. CRC Press, Inc., Boca Raton (2004)Google Scholar
18. 18.
Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(6), 611–626 (1989)
19. 19.
Suri, S.: The all-geodesic-furthest neighbors problem for simple polygons. In: Proc. 3rd Annu. Sympos. Comput. Geom. (SoCG), pp. 64–75 (1987)Google Scholar
20. 20.
Zalgaller, V.A.: An isoperimetric problem for tetrahedra. Journal of Mathematical Sciences 140(4), 511–527 (2007)

## Authors and Affiliations

• Sang Won Bae
• 1
• Matias Korman
• 2
• Yoshio Okamoto
• 3
1. 1.Department of Computer ScienceKyonggi UniversityKorea
2. 2.Computer Science DepartmentUniversité Libre de Bruxelles (ULB)Belgium
3. 3.Graduate School of Information Science and EngineeringTokyo Institute of TechnologyTokyoJapan