Abstract
In this paper, we consider the problem of choosing disks (that we can think of as corresponding to wireless sensors) so that given a set of input points in the plane, there exists no path between any pair of these points that is not intercepted by some disk. We try to achieve this separation using a minimum number of a given set of unit disks. We show that a constant factor approximation to this problem can be found in polynomial time using a greedy algorithm. To the best of our knowledge we are the first to study this optimization problem.
This material is based upon work supported by the National Science Foundation under Grant No. 0915543.
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References
Agarwal, P.K., Sharir, M.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1998)
Alt, H., Cabello, S., Giannopoulos, P., Knauer, C.: Minimum cell connection and separation in line segment arrangements (2011)(manuscript), http://arxiv.org/abs/1104.4618
Aronov, B., Ezra, E., Sharir, M.: Small-size epsilon-nets for axis-parallel rectangles and boxes. SIAM J. Comput. 39(7), 3248–3282 (2010)
Balister, P., Zheng, Z., Kumar, S., Sinha, P.: Trap coverage: Allowing coverage holes of bounded diameter in wireless sensor networks. In: Proc. of IEEE INFOCOM, Rio de Janeiro (2009)
Bereg, S., Kirkpatrick, D.: Approximating Barrier Resilience in Wireless Sensor Networks. In: Dolev, S. (ed.) ALGOSENSORS 2009. LNCS, vol. 5804, pp. 29–40. Springer, Heidelberg (2009)
Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete & Computational Geometry 14(4), 463–479 (1995)
Cabello, S.: Personal Communication (May 2011)
Chalermsook, P., Chuzhoy, J.: Maximum independent set of rectangles. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, pp. 892–901. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Chan, T.M., Har- Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. In: Proc. Symposium on Computational Geometry, SCG 2009, pp. 333–340 (2009)
Clarkson, K.L., Varadarajan, K.: Improved approximation algorithms for geometric set cover. In: Proc. Symposium on Computational Geometry, SCG 2005, pp. 135–141 (2005)
Erickson, J., Har- Peled, S.: Optimally cutting a surface into a disk. Discrete & Computational Geometry 31(1), 37–59 (2004)
Fox, J., Pach, J.: Computing the independence number of intersection graphs. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 (2011)
Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. Journal of Algorithms 50(1), 49–61 (2004)
Gibson, M., Kanade, G., Varadarajan, K.: On isolating points using disks (2011) (manuscript), http://arxiv.org/abs/1104.5043v1
Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1, 59–71 (1986)
Kumar, S., Lai, T. H., Arora, A.: Barrier coverage with wireless sensors. In: MobiCom 2005: Proceedings of the 11th Annual International Conference on Mobile Computing and Networking, pp. 284–298. ACM, New York (2005)
Mustafa, N.H., Ray, S.: PTAS for geometric hitting set problems via local search. In: Proc. Symposium on Computational Geometry, SCG 2009, pp. 17–22 (2009)
Reif, J.: Minimum s-t cut of a planar undirected network in o(n log2 n) time. SIAM Journal on Computing 12, 71–81 (1983)
Sankararaman, S., Efrat, A., Ramasubramanian, S., Taheri, J.: Scheduling sensors for guaranteed sparse coverage (2009) (manuscript), http://arxiv.org/abs/0911.4332
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Gibson, M., Kanade, G., Varadarajan, K. (2011). On Isolating Points Using Disks. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_6
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DOI: https://doi.org/10.1007/978-3-642-23719-5_6
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