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

DOI: 10.1007/978-3-319-57586-5_17

Part of the Lecture Notes in Computer Science book series (LNCS, volume 10236)
Cite this paper as:
Dobrev S. et al. (2017) Weak Coverage of a Rectangular Barrier. In: Fotakis D., Pagourtzis A., Paschos V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science, vol 10236. Springer, Cham

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.

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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