Weak Coverage of a Rectangular Barrier

  • Stefan Dobrev
  • Evangelos Kranakis
  • Danny Krizanc
  • Manuel Lafond
  • Jan Maňuch
  • Lata Narayanan
  • Jaroslav Opatrny
  • Sunil Shende
  • Ladislav Stacho
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)

Abstract

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak barrier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (MinMax). We give an \(O(n^{3/2})\) time algorithm for the MinNum problem for sensors of diameter 1 that are initially placed at integer positions; in contrast we show that the problem is NP-hard even for sensors of diameter 2 that are initially placed at integer positions. We show that the MinSum problem is solvable in \(O(n \log n)\) time for homogeneous range sensors in arbitrary initial positions for the Manhattan metric, while it is NP-hard for heterogeneous sensor ranges for both Manhattan and Euclidean metrics. Finally, we prove that even very restricted homogeneous versions of the MinMax problem are NP-hard.

References

  1. 1.
    Andrews, A.M., Wang, H.: Minimizing the aggregate movements for interval coverage. Algorithmica 78(1), 47–85 (2017). doi:10.1007/s00453-016-0153-8 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Balister, P., Bollobas, B., Sarkar, A., Kumar, S.: Reliable density estimates for coverage and connectivity in thin strips of finite length. In: ACM International Conference on Mobile Computing and Networking, pp. 75–86 (2007)Google Scholar
  3. 3.
    Ban, D., Jiang, J., Yang, W., Dou, W., Yi, H.: Strong \(k\)-barrier coverage with mobile sensors. In: Proceedings of International Wireless Communications and Mobile Computing Conference, pp. 68–72 (2010)Google Scholar
  4. 4.
    Berman, P., Karpinski, M.: On some tighter inapproximability results (extended abstract). In: Wiedermann, J., Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, pp. 200–209. Springer, Heidelberg (1999). doi:10.1007/3-540-48523-6_17 CrossRefGoogle Scholar
  5. 5.
    Bhattacharya, B., Burmester, M., Hu, Y., Kranakis, E., Shi, Q., Wiese, A.: Optimal movement of mobile sensors for barrier coverage of a planar region. TCS 410(52), 5515–5528 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chen, D.Z., Gu, Y., Li, J., Wang, H.: Algorithms on minimizing the maximum sensor movement for barrier coverage of a linear domain. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 177–188. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31155-0_16 CrossRefGoogle Scholar
  7. 7.
    Czyzowicz, J., et al.: On minimizing the maximum sensor movement for barrier coverage of a line segment. In: Ruiz, P.M., Garcia-Luna-Aceves, J.J. (eds.) ADHOC-NOW 2009. LNCS, vol. 5793, pp. 194–212. Springer, Heidelberg (2009). doi:10.1007/978-3-642-04383-3_15 CrossRefGoogle Scholar
  8. 8.
    Czyzowicz, J., et al.: On minimizing the sum of sensor movements for barrier coverage of a line segment. In: Nikolaidis, I., Wu, K. (eds.) ADHOC-NOW 2010. LNCS, vol. 6288, pp. 29–42. Springer, Heidelberg (2010). doi:10.1007/978-3-642-14785-2_3 CrossRefGoogle Scholar
  9. 9.
    Dobrev, S., Durocher, S., Eftekhari, M., Georgiou, K., Kranakis, E., Krizanc, D., Narayanan, L., Opatrny, J., Shende, S., Urrutia, J.: Complexity of barrier coverage with relocatable sensors in the plane. Theoret. Comput. Sci. 579, 64–73 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Eftekhari, M., Flocchini, P., Narayanan, L., Opatrny, J., Santoro, N.: Distributed barrier coverage with relocatable sensors. In: Halldórsson, M.M. (ed.) SIROCCO 2014. LNCS, vol. 8576, pp. 235–249. Springer, Cham (2014). doi:10.1007/978-3-319-09620-9_19 Google Scholar
  11. 11.
    Eftekhari, M., Kranakis, E., Krizanc, D., Morales-Ponce, O., Narayanan, L., Opatrny, J., Shende, S.: Distributed algorithms for barrier coverage using relocatable sensors. Distrib. Comput. 29(5), 361–376 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Eftekhari, M., Narayanan, L., Opatrny, J.: On multi-round sensor deployment for barrier coverage. In: Proceedings of 10th IEEE International Conference on Mobile Ad-hoc and Sensor Systems (IEEE MASS), pp. 310–318 (2013)Google Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman & Co., New York (1979)MATHGoogle Scholar
  14. 14.
    Habib, M.: Stochastic barrier coverage in wireless sensor networks based on distributed learning automata. Comput. Commun. 55, 51–61 (2015)CrossRefGoogle Scholar
  15. 15.
    Kranakis, E., Krizanc, D., Luccio, F.L., Smith, B.: Maintaining intruder detection capability in a rectangular domain with sensors. In: Bose, P., Gąsieniec, L.A., Römer, K., Wattenhofer, R. (eds.) ALGOSENSORS 2015. LNCS, vol. 9536, pp. 27–40. Springer, Cham (2015). doi:10.1007/978-3-319-28472-9_3 CrossRefGoogle Scholar
  16. 16.
    Kumar, S., Lai, T.H., Arora, A.: Barrier coverage with wireless sensors. In: Proceedings of the 11th Annual International Conference on Mobile Computing and Networking, pp. 284–298 (2005)Google Scholar
  17. 17.
    Li, L., Zhang, B., Shen, X., Zheng, J., Yao, Z.: A study on the weak barrier coverage problem in wireless sensor networks. Comput. Netw. 55, 711–721 (2011)CrossRefMATHGoogle Scholar
  18. 18.
    Mehrandish, M., Narayanan, L., Opatrny, J.: Minimizing the number of sensors moved on line barriers. In: Proceedings of IEEE WCNC, pp. 653–658 (2011)Google Scholar
  19. 19.
    Yan, G., Qiao, D.: Multi-round sensor deployment for guaranteed barrier coverage. In: Proceedings of IEEE INFOCOM 2010, pp. 2462–2470 (2010)Google Scholar
  20. 20.
    Dobrev, S., Kranakis, E., Krizanc, D., Lafond, M., Manuch, J., Narayanan, L., Opatrny, J., Stacho, L.: Weak coverage of a rectangular barrier. arXiv 1701.07294 (2017)

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Stefan Dobrev
    • 1
  • Evangelos Kranakis
    • 2
  • Danny Krizanc
    • 3
  • Manuel Lafond
    • 4
  • Jan Maňuch
    • 5
  • Lata Narayanan
    • 6
  • Jaroslav Opatrny
    • 6
  • Sunil Shende
    • 7
  • Ladislav Stacho
    • 8
  1. 1.Institute of MathematicsSlovak Academy of SciencesBratislavaSlovakia
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA
  4. 4.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  5. 5.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  6. 6.Department of Computer Science and Software EngineeringConcordia UniversityMontrealCanada
  7. 7.Department of Computer ScienceRutgers UniversityCamdenUSA
  8. 8.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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