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A Faster Exact-Counting Protocol for Anonymous Dynamic Networks

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Abstract

We study the problem of Counting the number of nodes in Anonymous Dynamic Network: nodes do not have identifiers and the network topology changes frequently. Counting is a fundamental task in distributed computing, for instance, to decide termination. Knowing what is the cost of anonymity is of paramount importance to understand what is feasible for future generations of Dynamic Networks, where the lack of nodes identifiers might facilitate mass production. Previous upper bounds to compute the exact network size are double-exponential. Strikingly, only linear complexity lower bounds are known, leaving open the question of whether efficient Counting protocols for Anonymous Dynamic Networks exist or not. In this work, we achieve an exponential speedup presenting Incremental Counting (IC), a distributed Counting protocol for Anonymous Dynamic Networks that has exponential time complexity and computes the exact size of the system. We complement the theoretical study evaluating IC experimentally. We tested a variety of network topologies that may appear in practice, including extremal cases such as trees, paths, and continuously changing topologies. Our simulations showed that IC is polynomial for all the inputs tested, paving the way to use it in practical applications where the network topology is predictable.

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Notes

  1. It is important to notice that the concept of energy is used for illustration purposes only. That is, what nodes actually exchange is just a number. This concept has nothing to do with the energy consumed by nodes.

  2. Throughout the paper, \(\log \) means logarithm base2, unless otherwise stated.

  3. The time bound proved in Theorem 1 holds only for \(n\ge 4\). Hence, we start with an estimate of 4. Nevertheless, Lemmas 3 and 4 prove correctness for any value of k. Therefore, the algorithm is correct for any \(k\le \), \(n>1\).

  4. The time bound proved in Theorem 1 holds only for \(c>\log 5\). Hence, we specify such value in the algorithm. Nevertheless, Lemmas 3 and 4 prove correctness for any value of \(c>1\). Therefore, the algorithm is correct for any \(c>1\).

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Acknowledgements

We thank Arnaud Casteigts for introducing the model to us, and Antonio Fernández Anta for useful discussions. This research was partially supported by ANR Project DISPLEXITY (ANR-11-BS02-014).

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Authors and Affiliations

  1. Department of Computer Science, Kean University, Union, NJ, USA

    Maitri Chakraborty

  2. LABRI, INP, University of Bordeaux, Bordeaux, France

    Alessia Milani

  3. Computer Science Department, Pace University, New York, NY, USA

    Miguel A. Mosteiro

Authors
  1. Maitri Chakraborty
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  2. Alessia Milani
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  3. Miguel A. Mosteiro
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Corresponding author

Correspondence to Miguel A. Mosteiro.

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Preliminary versions of this work have appeared in [6, 22].

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Chakraborty, M., Milani, A. & Mosteiro, M.A. A Faster Exact-Counting Protocol for Anonymous Dynamic Networks. Algorithmica 80, 3023–3049 (2018). https://doi.org/10.1007/s00453-017-0367-4

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