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Improved Spanners in Networks with Symmetric Directional Antennas

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Book cover Algorithms for Sensor Systems (ALGOSENSORS 2014)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 8847))

Abstract

Consider a set \(S\) of sensors in a plane, each equipped with a directional antenna of beamwidth \(\frac{\pi }{2}\) and radius \(r\in O(1)\). The antennas are symmetric, i.e. for two sensors \(u\) and \(v\) to communicate, \(u\) has to be covered by the antenna of \(v\) and vice versa. Assuming the Unit Disc Graph (\(\mathrm{UDG}\)) of \(S\) is connected, the problem we attack is: How to orient the antennas so that the resulting connectivity graph is a \(k\)-spanner of the \(\mathrm{UDG}\), while minimizing the radius used?

We provide two results: (a) \(7\)-spanner using radius \(33\), and (b) \(5\)-spanner, still using \(O(1)\) radius. This significantly improves the previous state of the art (\(8\)-spanner), even improving upon the pre previous best result (\(6\)-spanner) for beamwidth \(\frac{2\pi }{3}\).

Supported by VEGA 2\0136\12.

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Notes

  1. 1.

    Recall that \(r(v)\) and \(l(v)\) are the counter-clockwise and clockwise rays of antenna \(v\), respectively.

  2. 2.

    It starts at least \(p\), due to padding; the relevant place to look is at distance \(14+p\) from \(c(C_i)\), as that it according to Lemma 5 the furthest place in \(D(C_i)\) containing any vertices.

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Author information

Authors and Affiliations

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

    Stefan Dobrev

  2. Department of Computer Science, Comenius University, Bratislava, Slovakia

    Milan Plžík

Authors
  1. Stefan Dobrev
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  2. Milan Plžík
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Corresponding author

Correspondence to Stefan Dobrev .

Editor information

Editors and Affiliations

  1. Stony Brook University, Stonybrook, New York, USA

    Jie Gao

  2. University of Arizona, Tucson, Arizona, USA

    Alon Efrat

  3. TU Braunschweig, Braunschweig, Niedersachsen, Germany

    Sándor P. Fekete

  4. Rutgers University, New Brunswick, New Jersey, USA

    Yanyong Zhang

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Dobrev, S., Plžík, M. (2015). Improved Spanners in Networks with Symmetric Directional Antennas. In: Gao, J., Efrat, A., Fekete, S., Zhang, Y. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2014. Lecture Notes in Computer Science(), vol 8847. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46018-4_7

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  • DOI: https://doi.org/10.1007/978-3-662-46018-4_7

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-46017-7

  • Online ISBN: 978-3-662-46018-4

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