Abstract
Given n points in a circular region C in the plane, we study the problem of moving these points to the boundary of C to form a regular n-gon such that the maximum of the Euclidean distances traveled by the points is minimized. These problems find applications in mobile sensor barrier coverage of wireless sensor networks. The problem further has two versions: the decision version and optimization version. In this paper, we present an O(nlog2 n) time algorithm for the decision version and an O(nlog3 n) time algorithm for the optimization version. The previously best algorithms for these two problem versions take O(n 3.5) time and O(n 3.5logn) time, respectively. A by-product of our techniques is an algorithm for dynamically maintaining the maximum matching of a circular convex bipartite graph; our algorithm performs each vertex insertion or deletion on the graph in O(log2 n) time. This result may be interesting in its own right.
Chen’s research was supported in part by NSF under Grant CCF-0916606. Work by Tan was partially supported by Grant-in-Aid (MEXT/JPSP KAKENHI 23500024) for Scientific Research from Japan Society for the Promotion of Science.
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Bhattacharya, B., Burmester, B., Hu, Y., Kranakis, E., Shi, Q., Wiese, A.: Optimal movement of mobile sensors for barrier coverage of a planar region. Theoretical Computer Science 410(52), 5515–5528 (2009)
Bremner, D., Chan, T., Demaine, E., Erickson, J., Hurtado, F., Iacono, J., Langerman, S., Taslakian, P.: Necklaces, convolutions, and X + Y. In: Proc. of the 14th conference on Annual European Symposium on Algorithms, pp. 160–171 (2006)
Brodal, G.S., Georgiadis, L., Hansen, K.A., Katriel, I.: Dynamic Matchings in Convex Bipartite Graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 406–417. Springer, Heidelberg (2007)
Chang, M., Tang, C., Lee, R.: Solving the Euclidean bottleneck matching problem by k-relative neighborhood graphs. Algorithmica 8, 177–194 (1992)
Chen, A., Kumar, S., Lai, T.: Designing localized algorithms for barrier coverage. In: Proc. of the 13th Annual ACM International Conference on Mobile Computing and Networking, pp. 63–73 (2007)
Chen, D., Tan, X., Wang, H., Wu, G.: Optimal point movement for covering circular regions. arXiv:1107.1012v1 (2012)
Chen, D., Wang, C., Wang, H.: Representing a functional curve by curves with fewer peaks. Discrete and Computational Geometry 46(2), 334–360 (2011)
Cole, R.: Slowing down sorting networks to obtain faster sorting algorithms. Journal of the ACM 34(1), 200–208 (1987)
Cole, R., Salowe, J., Steiger, W., Szemerédi, E.: An optimal-time algorithm for slope selection. SIAM Journal on Computing 18(4), 792–810 (1989)
Efrat, A., Itai, A., Katz, M.: Geometry helps in bottleneck matching and related problems. Algorithmica 31(1), 1–28 (2001)
Efrat, A., Katz, M.: Computing Euclidean bottleneck matchings in higher dimensions. Information Processing Letters 75, 169–174 (2000)
Gabow, H., Tarjan, R.: A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences 30, 209–221 (1985)
Lipski Jr., W., Preparata, F.P.: Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems. Acta Informatica 15(4), 329–346 (1981)
Kumar, S., Lai, T., Arora, A.: Barrier coverage with wireless sensors. Wireless Networks 13(6), 817–834 (2007)
Liang, Y., Blum, N.: Circular convex bipartite graphs: Maximum matching and Hamiltonian circuits. Information Processing Letters 56, 215–219 (1995)
Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM 30(4), 852–865 (1983)
Steiner, G., Yeomans, J.: A linear time algorithm for maximum matchings in convex, bipartite graphs. Computers and Mathematics with Applications 31(2), 91–96 (1996)
Tan, X., Wu, G.: New Algorithms for Barrier Coverage with Mobile Sensors. In: Lee, D.-T., Chen, D.Z., Ying, S. (eds.) FAW 2010. LNCS, vol. 6213, pp. 327–338. Springer, Heidelberg (2010)
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Chen, D.Z., Tan, X., Wang, H., Wu, G. (2012). Optimal Point Movement for Covering Circular Regions. In: Chao, KM., Hsu, Ts., Lee, DT. (eds) Algorithms and Computation. ISAAC 2012. Lecture Notes in Computer Science, vol 7676. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35261-4_36
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DOI: https://doi.org/10.1007/978-3-642-35261-4_36
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