Abstract
Given a set P of n points and a straight line L, we study three important variations of minimum enclosing circle problem. The first problem is on computing k circles of minimum (common) radius with centers on L which can cover the members in P. We propose three algorithms for this problem. The first one runs in O(nklogn) time and O(n) space. The second one runs in O(nk + k 2log3 n) time and O(nlogn) space assuming that the points are sorted along L, and is efficient where k < < n. The third one is based on parametric search and it runs in O(nlogn + klog4 n) time. The next one is on computing the minimum radius circle centered on L that can enclose at least k points. The time and space complexities of the proposed algorithm are O(nk) and O(n) respectively. Finally, we study the situation where the points are associated with k colors, and the objective is to find a minimum radius circle with center on L such that at least one point of each color lies inside it. We propose an O(nlogn) time algorithm for this problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Sacriston, V.: The farthest color voronoi diagram and related problems. In: 17th European Workshop on Computational Geometry (2001)
Agarwal, P.K., Sharir, M., Welzl, E.: The discrete 2-center problem. In: 13th Annual Symposium on Computational Geometry, pp. 147–155 (1997)
Alt, H., Arkin, E.M., Bronnimann, H., Erickson, J., Fekete, S.P., Knauer, C., Lenchner, J., Mitchell, J.S.B., Whittlesey, K.: Minimum-cost Coverage of point sets by disks. In: Proc. 22nd Annual ACM Symposium on Computational Geometry, pp. 449–458 (2006)
Brass, P., Knauer, C., H.-S. Na, C.-S. Shin, Vigneron, A.: Computing k-centers on a line Technical Report CoRR abs/0902.3282 (2009)
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)
Chan, T.M.: More planar two-center algorithms. Computational Geometry: Theory and Applications 13, 189–198 (1999)
Cole, R., Salowe, J., Steiger, W., Szemeredi, E.: An optimal-time algorithm for slope selection. SIAM Journal on Computing 18, 792–810 (1989)
Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. J. ACM 34, 200–208 (1987)
Gupta, U.I., Lee, D.T., Leung, J.Y.-T.: Efficient algorithms for interval graphs and circular-arc graphs. Networks 12, 459–467 (1982)
Hwang, R.Z., Chang, R.C., Lee, R.C.T.: The searching over separators strategy To solve some NP-hard problems in subexponential time. Algorithmica 9, 398–423 (1993)
Huttenlocher, D.P., Kedem, K., Sharir, M.: The upper envelope of Voronoi surfaces and its applications. Discrete Computational Geometry 9, 267–291 (1993)
Hurtado, F., Sacristan, V., Toussaint, G.: Facility location problems with constraints. Studies in Locational Analysis 15, 17–35 (2000)
Hochbaum, D.S., Shmoys, D.: A best possible heuristic for the k- center problem. Mathematics of Operations Research 10, 180–184 (1985)
Karmakar, A., Roy, S., Das, S.: Fast computation of smallest enclosing circle with center on a query line segment. Information Processing Letters 108, 343–346 (2008)
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)
Marchetti-Spaccamela, A.: The p-center problem in the plane is NP-complete. In: Proc. 19th Allerton Conf. on Communication, Control and Computing, pp. 31–40 (1981)
Matousek, J.: Randomized optimal algorithm for slope selection. Information Processing Letters 39, 183–187 (1991)
Megiddo, N.: Linear-time algorithms for linear programming in R 3 and related problems. SIAM Journal Comput. 12, 759–776 (1983)
Olariu, S., Schwing, J.L., Zhang, J.: Optimal parallel algorithms for problems modeled by a family of intervals. IEEE Transactions on Parallel and Distributed Systems 3, 364–374 (1992)
Plesnik, J.: A heuristic for the p-center problem in graphs. Discrete Applied Mathematics 17, 263–268 (1987)
Sharir, M.: A near-linear algorithm for the planar 2-center problem. Discrete Computational Geometry 18, 125–134 (1997)
Roy, S., Karmakar, A., Das, S., Nandy, S.C.: Constrained minimum enclosing circle with center on a query line Segment. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 765–776. Springer, Heidelberg (2006)
Vahrenhold, J.: Line-segment intersection made in-place. Computational Geometry: Theory and Applications 38, 213–230 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Karmakar, A., Das, S., Nandy, S.C., Bhattacharya, B.K. (2010). Some Variations on Constrained Minimum Enclosing Circle Problem. In: Wu, W., Daescu, O. (eds) Combinatorial Optimization and Applications. COCOA 2010. Lecture Notes in Computer Science, vol 6508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17458-2_29
Download citation
DOI: https://doi.org/10.1007/978-3-642-17458-2_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17457-5
Online ISBN: 978-3-642-17458-2
eBook Packages: Computer ScienceComputer Science (R0)