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Labeling Schemes for Deterministic Radio Multi-broadcast

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Graph-Theoretic Concepts in Computer Science (WG 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12911))

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Abstract

We consider the multi-broadcast problem in arbitrary connected radio networks consisting of n nodes. There are k designated source nodes for some fixed \(k \in \{1,\ldots ,n\}\), and each source node has a piece of information that it wants to share with all nodes in the network. We set out to determine the shortest possible labels so that multi-broadcast can be solved deterministically in the labeled radio network by some deterministic distributed algorithm. First, we show that every radio network G with maximum degree \(\varDelta \) can be labeled using \(O(\min \{\log k,\log \varDelta \})\)-bit labels in such a way that multi-broadcast with k sources can be accomplished. This bound is tight for certain network topologies (e.g., complete graphs), but there are networks where significantly shorter labels are sufficient, e.g., we show how to construct a tree with maximum degree \(\varTheta (\sqrt{n})\) in which gossiping (i.e., multi-broadcast with n sources) can be solved after labeling each node with O(1) bits. For all trees, we provide a labeling scheme and algorithm that will solve gossiping, and, we prove an impossibility result that demonstrates that our labeling scheme is optimal for gossiping in each tree. In particular, we prove that \(\varTheta (\log D(G))\)-bit labels are necessary and sufficient in each tree G, where D(G) denotes the distinguishing number of G.

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Acknowledgments

Avery Miller acknowledges the support of NSERC, Discovery Grant RGPIN-2017-05936.

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  1. University of Manitoba, Winnipeg, MB, Canada

    Colin Krisko & Avery Miller

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  1. Colin Krisko
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Correspondence to Avery Miller .

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Editors and Affiliations

  1. University of Warsaw, Warsaw, Poland

    Łukasz Kowalik

  2. University of Warsaw, Warsaw, Poland

    Michał Pilipczuk

  3. Warsaw University of Technology, Warsaw, Poland

    Paweł Rzążewski

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Krisko, C., Miller, A. (2021). Labeling Schemes for Deterministic Radio Multi-broadcast. In: Kowalik, Ł., Pilipczuk, M., Rzążewski, P. (eds) Graph-Theoretic Concepts in Computer Science. WG 2021. Lecture Notes in Computer Science(), vol 12911. Springer, Cham. https://doi.org/10.1007/978-3-030-86838-3_29

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