# Optimal Algorithms for Constrained 1-Center Problems

• Luis Barba
• Prosenjit Bose
• Stefan Langerman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

## Abstract

We address the following problem: Given two subsets Γ and Φ of the plane, find the minimum enclosing circle of Γ whose center is constrained to lie on Φ. We first study the case when Γ is a set of n points and Φ is either a set of points, a set of segments (lines) or a simple polygon. We propose several algorithms, the first solves the problem when Φ is a set of m segments (or m points) in expected Θ((n + m)logω) time, where ω =  min {n, m}. Surprisingly, when Φ is a simple m-gon, we can improve the expected running time to Θ(m + nlogn). Moreover, if Γ is the set of vertices of a convex n-gon and Φ is a simple m-gon, we can solve the problem in expected Θ(n + m) time. We provide matching lower bounds in the algebraic computation tree model for all the algorithms presented in this paper. While proving these results, we obtained a Ω(n logm) lower bound for the following problem: Given two sets A and B in ℝ of sizes m and n, respectively, decide if A is a subset of B.

## Keywords

minimum enclosing circle facility location problems

## References

1. 1.
Aggarwal, A., Guibas, L., Saxe, J., Shor, P.: A linear time algorithm for computing the Voronoi diagram of a convex polygon. In: Proceedings of STOC, pp. 39–45. ACM, New York (1987)Google Scholar
2. 2.
Ben-Or, M.: Lower bounds for algebraic computation trees. In: Proceedings of STOC, pp. 80–86. ACM, New York (1983)Google Scholar
3. 3.
Bose, P., Langerman, S., Roy, S.: Smallest enclosing circle centered on a query line segment. In: Proceedings of CCCG, pp. 167–170 (2008)Google Scholar
4. 4.
Bose, P., Toussaint, G.: Computing the constrained Euclidean, geodesic and link centre of a simple polygon with applications. In: Proceedings of CGI, pp. 102–111 (1996)Google Scholar
5. 5.
Bose, P., Wang, Q.: Facility location constrained to a polygonal domain. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 153–164. Springer, Heidelberg (2002)
6. 6.
Chan, T.M.: Geometric applications of a randomized optimization technique. Discrete and Computational Geometry 22, 547–567 (1999)
7. 7.
Chazelle, B., Edelsbrunner, H., Guibas, L., Sharir, M.: Diameter, width, closest line pair, and parametric searching. DCG 10, 183–196 (1993)
8. 8.
Hurtado, F., Sacristan, V., Toussaint, G.: Some constrained minimax and maximim location problems. Studies in Locational Analysis 15, 17–35 (2000)
9. 9.
Joe, B., Simpson, R.B.: Corrections to Lee’s visibility polygon algorithm. BIT Numerical Mathematics 27, 458–473 (1987)
10. 10.
Lee, D.T.: Farthest neighbor Voronoi diagrams and applications. Report 80-11-FC-04, Dept. Elect. Engrg. Comput. Sci. (1980)Google Scholar
11. 11.
Lee, D.T.: On finding the convex hull of a simple polygon. International Journal of Parallel Programming 12(2), 87–98 (1983)
12. 12.
Matousek, J.: Computing the center of planar point sets. Discrete and Computational Geometry 6, 221 (1991)Google Scholar
13. 13.
Matoušek, J.: Construction of epsilon nets. In: Proceedings of SCG, pp. 1–10. ACM, New York (1989)Google Scholar
14. 14.
Megiddo, N.: Linear-time algorithms for linear programming in ℝ3 and related problems. SIAM J. Comput. 12(4), 759–776 (1983)
15. 15.
Preparata, F.: Minimum spanning circle. In: Preparata, F.P. (ed.) Steps in Computational Geometry. University of Illinois (1977)Google Scholar
16. 16.
Shamos, M., Hoey, D.: Closest-point problems. In: Proceedings of FOCS, pp. 151–162. IEEE Computer Society, Washington, DC (1975)Google Scholar
17. 17.
Sylvester, J.J.: A Question in the Geometry of Situation. Quarterly Journal of Pure and Applied Mathematics 1 (1857)Google Scholar
18. 18.
Yao, A.C.-C.: Decision tree complexity and Betti numbers. In: Proceedings of STOC, pp. 615–624. ACM, New York (1994)Google Scholar

## Authors and Affiliations

• Luis Barba
• 1
• 2
• Prosenjit Bose
• 1
• Stefan Langerman
• 2