Algorithms for Covering Multiple Barriers

  • Shimin Li
  • Haitao WangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)


In this paper, we consider the problems for covering multiple intervals on a line. Given a set B of m line segments (called “barriers”) on a horizontal line L and another set S of n horizontal line segments of the same length in the plane, we want to move all segments of S to L so that their union covers all barriers and the maximum movement of all segments of S is minimized. Previously, an \(O(n^3\log n)\)-time algorithm was given for the problem but only for the special case \(m=1\). In this paper, we propose an \(O(n^2\log n\log \log n+nm\log m)\)-time algorithm for any m, which improves the previous work even for \(m=1\). We then consider a line-constrained version of the problem in which the segments of S are all initially on the line L. Previously, an \(O(n\log n)\)-time algorithm was known for the case \(m=1\). We present an algorithm of \(O((n+m)\log (n+ m))\) time for any m. These problems may have applications in mobile sensor barrier coverage in wireless sensor networks.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, A., Wang, H.: Minimizing the aggregate movements for interval coverage. Algorithmica 78, 47–85 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, D., Gu, Y., Li, J., Wang, H.: Algorithms on minimizing the maximum sensor movement for barrier coverage of a linear domain. Discrete and Computational Geometry 50, 374–408 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, D., Tan, X., Wang, H., Wu, G.: Optimal point movement for covering circular regions. Algorithmica 72, 379–399 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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
  6. 6.
    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
  7. 7.
    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. Theoretical Computer Science 579, 64–73 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Frederickson, G., Johnson, D.: Generalized selection and ranking: Sorted matrices. SIAM Journal on Computing 13(1), 14–30 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kumar, S., Lai, T., Arora, A.: Barrier coverage with wireless sensors. In: Proc. of the 11th Annual International Conference on Mobile Computing and Networking (MobiCom), pp. 284–298 (2005)Google Scholar
  10. 10.
    Li, S., Shen, H.: Minimizing the maximum sensor movement for barrier coverage in the plane. In: Proc. of the 2015 IEEE Conference on Computer Communications (INFOCOM), pp. 244–252 (2015)Google Scholar
  11. 11.
    Li, S., Wang, H.: Algorithms for covering multiple barriers. arXiv:1704.06870 (2017)
  12. 12.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. Journal of the ACM 30(4), 852–865 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mehrandish, M.: On Routing, Backbone Formation and Barrier Coverage in Wireless Ad Doc and Sensor Networks. Ph.D. thesis, Concordia University, Montreal, Quebec, Canada (2011)Google Scholar
  14. 14.
    Mehrandish, M., Narayanan, L., Opatrny, J.: Minimizing the number of sensors moved on line barriers. In: Proc. of IEEE Wireless Communications and Networking Conference (WCNC), pp. 653–658 (2011)Google Scholar
  15. 15.
    Wang, H., Zhang, X.: Minimizing the maximum moving cost of interval coverage. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 188–198. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-48971-0_17. Full version to appear in International Journal of Computational Geometry and Application (IJCGA)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUtah State UniversityLoganUSA

Personalised recommendations