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Collaborative Broadcast in \(\mathcal {O}(\log \log n)\) Rounds

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Algorithms for Sensor Systems (ALGOSENSORS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 11931))

Abstract

We consider the multihop broadcasting problem for n nodes placed uniformly at random in a disk and investigate the number of hops required to transmit a signal from the central node to all other nodes under three communication models: Unit-Disk-Graph (UDG), Signal-to-Noise-Ratio (SNR), and the wave superposition model of multiple input/multiple output (MIMO).

In the MIMO model, informed nodes cooperate to produce a stronger superposed signal. We do not consider the problem of transmitting a full message nor do we consider interference with other messages. In each round, the informed senders try to deliver to other nodes the required signal strength such that the received signal can be distinguished from the noise.

We assume a sufficiently high node density \(\rho = \varOmega (\log n)\) in order to launch the broadcasting process. In the unit-disk graph model, broadcasting takes \(\mathcal {O}(\sqrt{n/\rho })\) rounds. In the other models, we use an Expanding Disk Broadcasting Algorithm, where in a round only triggered nodes within a certain distance from the initiator node contribute to the broadcasting operation.

This algorithm achieves a broadcast in only \(\mathcal {O}\left( \frac{\log n}{\log \rho }\right) \) rounds in the SNR-model. Adapted to the MISO model, it broadcasts within \(\mathcal {O}(\log \log n - \log \log \rho )\) rounds. All bounds are asymptotically tight and hold with high probability, i.e. \(1- n^{-\mathcal {O}(1)}\).

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Acknowledgments

We like to thank the organizers of the Dagstuhl Seminar 17271, July 2–7, 2017, Foundations of Wireless Networking, where this research has begun and first results have been found. We would like to thank Alexander Leibold, who performed and checked the automated proofs and anonymous reviewers of a previous version for their detailed and valuable input. We would also like to thank Tigran Tonoyan, Magnús M. Halldórsson and Zvi Lotker for many fruitful discussions.

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Authors and Affiliations

  1. University of Freiburg, Georges-Köhler-Allee 51, 79110, Freiburg im Breisgau, Germany

    Christian Schindelhauer & Thomas Janson

  2. Technical University of Darmstadt, Hochschulstraße 10, 64289, Darmstadt, Germany

    Aditya Oak

Authors
  1. Christian Schindelhauer
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  2. Aditya Oak
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  3. Thomas Janson
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Correspondence to Christian Schindelhauer .

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Editors and Affiliations

  1. Paderborn University, Paderborn, Germany

    Falko Dressler

  2. Paderborn University, Paderborn, Germany

    Christian Scheideler

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Schindelhauer, C., Oak, A., Janson, T. (2019). Collaborative Broadcast in \(\mathcal {O}(\log \log n)\) Rounds. In: Dressler, F., Scheideler, C. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2019. Lecture Notes in Computer Science(), vol 11931. Springer, Cham. https://doi.org/10.1007/978-3-030-34405-4_7

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  • DOI: https://doi.org/10.1007/978-3-030-34405-4_7

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