ISAAC 2015: Algorithms and Computation pp 307-317

# The 2-Center Problem in a Simple Polygon

• Eunjin Oh
• Jean-Lou De Carufel
• Hee-Kap Ahn
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

## Abstract

The geodesic k-center problem in a simple polygon with n vertices consists in the following. Find k points, called centers, in the polygon to minimize the maximum geodesic distance from any point of the polygon to its closest center. In this paper, we focus on the case where $$k=2$$ and present an exact algorithm that returns an optimal geodesic 2-center in $$O(n^2\log ^2 n)$$ time.

## Keywords

Geodesic Distance Decision Algorithm Combinatorial Structure Simple Polygon Event Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 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: Arge, L., Pach, J. (eds.) 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), vol. 34, pp. 209–223. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2015)Google Scholar
2. 2.
Asano, T., Toussaint, G.T.: Computing geodesic center of a simple polygon. Technical Report SOCS-85.32, McGill University (1985)Google Scholar
3. 3.
Borgelt, M.G., Van Kreveld, M., Luo, J.: Geodesic disks and clustering in a simple polygon. Int. J. Comput. Geom. Appl. 21(06), 595–608 (2011)
4. 4.
Chan, T.M.: More planar two-center algorithms. Comput. Geom. 13(3), 189–198 (1999)
5. 5.
Chang, H.-C., Erickson, J., Xu, C.: Detecting weakly simple polygons. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2015), pp. 1655–1670. SIAM (2015)Google Scholar
6. 6.
Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms 21(3), 579–597 (1996)
7. 7.
Cole, R.: Parallel merge sort. SIAM J. Comput. 17(4), 770–785 (1988)
8. 8.
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(1–4), 209–233 (1987)
9. 9.
Hwang, R., Lee, R., Chang, R.: The slab dividing approach to solve the Euclidean p-center problem. Algorithmica 9(1), 1–22 (1993)
10. 10.
Kim, S.K., Shin, C.-S.: Efficient algorithms for two-center problems for a convex polygon. In: Du, D.-Z., Eades, P., Sharma, A.K., Lin, X., Estivill-Castro, V. (eds.) COCOON 2000. LNCS, vol. 1858, pp. 299–309. Springer, Heidelberg (2000)
11. 11.
Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983)
12. 12.
Megiddo, N.: On the ball spanned by balls. Discrete Comput. Geom. 4(1), 605–610 (1989)
13. 13.
Pollack, R., Sharir, M., Rote, G.: Computing the geodesic center of a simple polygon. Discrete Comput. Geom. 4(1), 611–626 (1989)
14. 14.
Vigan, I.: Packing and covering a polygon with geodesic disks. Technical report, CoRR abs/1311.6033 (2013)Google Scholar