Abstract
Given a set P of n points in the plane, the k-center problem is to find k congruent disks of minimum possible radius such that their union covers all the points in P. The 2-center problem is a special case of the k-center problem that has been extensively studied in the recent past [7, 20, 22]. In this paper, we consider a generalized version of the 2-center problem called proximity connected 2-center (PCTC) problem. In this problem, we are also given a parameter \(\delta \ge 0\) and we have the additional constraint that the distance between the centers of the disks should be at most \(\delta \). Note that when \(\delta =0\), the PCTC problem is reduced to the 1-center(minimum enclosing disk) problem and when \(\delta \) tends to infinity, it is reduced to the 2-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 [12], but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic \(O(n^2\log n)\) time algorithm for the problem.
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References
Agarwal, P.K., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33(2), 201–226 (2002)
Agarwal, P.K., Sharir, M.: Planar geometric location problems. Algorithmica 11(2), 185–195 (1994)
Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. (CSUR) 30(4), 412–458 (1998)
Arora, S.: Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. In: Proceedings 38th Annual Symposium on Foundations of Computer Science, pp. 554–563. IEEE, 20 October 1997
Chan, T.M.: More planar two-center algorithms. Comput. Geom. 13(3), 189–198 (1999)
Cho, K., Oh, E.: Optimal algorithm for the planar two-center problem. arXiv preprint arXiv:2007.08784, 17 July 2020
Choi, J., Ahn, H.K.: Efficient planar two-center algorithms. Comput. Geom. 2, 101768 (2021)
Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. J. ACM (JACM) 34(1), 200–208 (1987)
Drezner, Z.: The planar two-center and two-median problems. Transp. Sci. 18(4), 351–361 (1984)
Eppstein, D.: Faster construction of planar two-centers. In: Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 131–138 (1997)
Hershberger, J.: A faster algorithm for the two-center decision problem. Inf. Process. Lett. 47(1), 23–29 (1993)
Huang, C.H.: Some problems on radius-weighted model of packet radio network. Doctoral dissertation, Ph.D. Dissertation, Department of Computer Science, Tsing Hua University, Hsinchu, Taiwan (1992)
Huang, P.H., Tsai, Y.T., Tang, C.Y.: A near-quadratic algorithm for the alpha-connected two-center problem. J. Inf. Sci. Eng. 22(6), 1317 (2006)
Huang, P.H., Te Tsai, Y., Tang, C.Y.: A fast algorithm for the alpha-connected two-center decision problem. Inf. Process. Lett. 85(4), 205–10 (2003)
Katz, M.J., Sharir, M.: An expander-based approach to geometric optimization. SIAM J. Comput. 26(5), 1384–1408 (1997)
Gowda, I., Kirkpatrick, D., Lee, D., Naamad, A.: Dynamic voronoi diagrams. IEEE Trans. Inf. Theory 29(5), 724–731 (1983)
Megiddo, N.: Linear-time algorithms for linear programming in \(\mathbb{R}^3\) and related problems. SIAM J. Comput. 12(4), 759–776 (1983)
Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM (JACM) 30(4), 852–865 (1983)
Bhattacharya, B., Mozafari, A., Shermer, T.C.: An efficient algorithm for the proximity connected two center problem. arXiv preprint arXiv:2204.08754, 19 Apr 2022
Sharir, M.: A near-linear algorithm for the planar 2-center problem. Discret. Comput. Geom. 18(2), 125–34 (1997)
Toth, C.D., O’Rourke, J., Goodman, J.E. (eds.): Handbook of discrete and computational geometry. CRC Press, Boca Raton (2017)
Wang, H.: On the planar two-center problem and intersection hulls. arXiv preprint arXiv:2002.07945, 9 February 2020
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Bhattacharya, B., Mozafari, A., Shermer, T.C. (2022). An Efficient Algorithm for the Proximity Connected Two Center Problem. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_15
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