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Probabilistic Properties of Highly Connected Random Geometric Graphs

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Algorithms and Discrete Applied Mathematics (CALDAM 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10743))

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Abstract

In this paper, we study the probabilistic properties of reliable networks of minimum costs in d-dimensional Euclidean space. We study reliability in terms of k-edge-connectivity in graphs. We show that this problem fits into Yukich’s framework for Euclidean functionals for arbitrary k, dimension d and distant-power gradient p with \(p<d\). With this framework results on convergence and concentration of the value of optimal solutions of random inputs follow. These results are then extended to optimal k-edge-connected power assignment graphs, where we assign transmit power to vertices, and two vertices are connected if they both have sufficient transmit power. This variant models wireless networks. Finally, we devise a partitioning heuristic to find approximate solutions quickly, and we analyze its performance in the framework of smoothed analysis.

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Authors and Affiliations

  1. University of Twente, Enschede, The Netherlands

    Bodo Manthey & Victor M. J. J. Reijnders

Authors
  1. Bodo Manthey
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  2. Victor M. J. J. Reijnders
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Correspondence to Victor M. J. J. Reijnders .

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Editors and Affiliations

  1. Indian Institute of Technology Delhi, New Delhi, India

    B.S. Panda

  2. University of Calcutta, Kolkata, India

    Partha P. Goswami

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Manthey, B., Reijnders, V.M.J.J. (2018). Probabilistic Properties of Highly Connected Random Geometric Graphs. In: Panda, B., Goswami, P. (eds) Algorithms and Discrete Applied Mathematics. CALDAM 2018. Lecture Notes in Computer Science(), vol 10743. Springer, Cham. https://doi.org/10.1007/978-3-319-74180-2_5

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  • DOI: https://doi.org/10.1007/978-3-319-74180-2_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74179-6

  • Online ISBN: 978-3-319-74180-2

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