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Weak Coverage of a Rectangular Barrier

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Abstract

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. Each sensor can monitor a circular area of specific diameter around its position, called the sensor diameter. Sensors are required to move to final locations so that they can there 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 any sensor (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 the problem is shown to be NP-complete 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 the Manhattan metric and sensors of identical diameter (homogeneous sensors) in arbitrary initial positions, while it is NP-complete for heterogeneous sensors. Finally, we prove that even very restricted homogeneous versions of the MinMax problem are NP-complete.

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Authors and Affiliations

  1. Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovakia

    Stefan Dobrev

  2. School of Computer Science, Carleton University, Ottawa, Canada

    Evangelos Kranakis

  3. Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT, USA

    Danny Krizanc

  4. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada

    Manuel Lafond

  5. Department of Computer Science, University of British Columbia, Vancouver, Canada

    Ján Maňuch

  6. Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada

    Lata Narayanan & Jaroslav Opatrny

  7. Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada

    Ladislav Stacho

Authors
  1. Stefan Dobrev
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  2. Evangelos Kranakis
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  3. Danny Krizanc
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  4. Manuel Lafond
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  5. Ján Maňuch
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  6. Lata Narayanan
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  7. Jaroslav Opatrny
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  8. Ladislav Stacho
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Corresponding author

Correspondence to Jaroslav Opatrny.

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An extended abstract of this paper appeared in CIAC 2017 [11]. Research supported in part by NSERC grants, Canada, and a VEGA grant, Slovakia.

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Dobrev, S., Kranakis, E., Krizanc, D. et al. Weak Coverage of a Rectangular Barrier. Algorithmica 82, 721–746 (2020). https://doi.org/10.1007/s00453-019-00611-7

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  • DOI: https://doi.org/10.1007/s00453-019-00611-7

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