On Isolating Points Using Disks

  • Matt Gibson
  • Gaurav Kanade
  • Kasturi Varadarajan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)

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.

Keywords

Wireless Sensor Network Approximation Algorithm Unit Disk Greedy Algorithm Intersection Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, P.K., Sharir, M.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Aronov, B., Ezra, E., Sharir, M.: Small-size epsilon-nets for axis-parallel rectangles and boxes. SIAM J. Comput. 39(7), 3248–3282 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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)CrossRefGoogle Scholar
  6. 6.
    Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete & Computational Geometry 14(4), 463–479 (1995)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cabello, S.: Personal Communication (May 2011)Google Scholar
  8. 8.
    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)Google Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    Clarkson, K.L., Varadarajan, K.: Improved approximation algorithms for geometric set cover. In: Proc. Symposium on Computational Geometry, SCG 2005, pp. 135–141 (2005)Google Scholar
  11. 11.
    Erickson, J., Har- Peled, S.: Optimally cutting a surface into a disk. Discrete & Computational Geometry 31(1), 37–59 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fox, J., Pach, J.: Computing the independence number of intersection graphs. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, SODA 2011 (2011)Google Scholar
  13. 13.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Multiway cuts in node weighted graphs. Journal of Algorithms 50(1), 49–61 (2004)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Gibson, M., Kanade, G., Varadarajan, K.: On isolating points using disks (2011) (manuscript), http://arxiv.org/abs/1104.5043v1
  15. 15.
    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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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)Google Scholar
  17. 17.
    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)Google Scholar
  18. 18.
    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)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Sankararaman, S., Efrat, A., Ramasubramanian, S., Taheri, J.: Scheduling sensors for guaranteed sparse coverage (2009) (manuscript), http://arxiv.org/abs/0911.4332

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Matt Gibson
    • 1
  • Gaurav Kanade
    • 2
  • Kasturi Varadarajan
    • 2
  1. 1.Department of Electrical and Computer EngineeringThe University of IowaIowa CityUSA
  2. 2.Department of Computer ScienceThe University of IowaIowa CityUSA

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