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Distributed backup placement

for networks of bounded neighborhood independence and networks of bounded arboricity

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Abstract

We consider the Backup Placement problem in networks in the \(\mathcal {CONGEST}\) distributed setting. Given a network graph \(G = (V,E)\), the goal of each vertex \(v \in V\) is selecting a neighbor, such that the maximum number of vertices in V that select the same vertex is minimized. The backup placement problem was introduced by Halldorsson, Kohler, Patt-Shamir, and Rawitz, who obtained in 2015 an \(O(\log n/ \log \log n)\) approximation with randomized polylogarithmic time. Their algorithm remained state-of-the-art for general graphs, as well as for specific graph topologies. In the current paper, we obtain significantly improved algorithms for various graph topologies. Specifically, we show that O(1)-approximation to optimal backup placement can be computed deterministically in O(1) rounds (and even just one round) in wireless networks, certain social networks, claw-free graphs, and, more precisely, in any graph with neighborhood independence bounded by a constant. We also consider graphs such as trees, forests, planar graphs and, more precisely, graphs of constant arboricity. For such graphs, we obtain constant approximation to optimal backup placement in \(O(\log n)\) deterministic rounds. Clearly, our constant-time algorithms for graphs with constant neighborhood independence are asymptotically optimal. Moreover, we show that our algorithms for graphs with constant arboricity are not far from optimal as well by proving several lower bounds. Specifically, in unoriented trees, optimal backup placement requires \(\Omega (\log n)\) time and polylogarithmic-approximate backup placement requires \(\Omega (\sqrt{\log n / \log \log n})\) time. These lower bounds are applicable in particular to graphs of constant arboricity.

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Notes

  1. Note that in this work we focus on the unweighted version of the problem. Some works also consider weighted cases; however, they are out of the scope of this work.

  2. Note that from a point of view of a vertex, all of its neighbors are the same in terms of load assignment, even with unique IDs. Thus, additional computations have to be performed to select neighbors for assignments and break this type of symmetry.

  3. Arboricity is the minimum number of forests that the graph edges can be partitioned into or the maximum ratio of edges to nodes in any subgraph. The arboricity of a graph is a measure for its sparsity. Sparse graphs have low arboricity.

  4. A graph is growth-bounded (GBG) or independence-bounded (BIG) if the number of independent nodes in a node’s r-neighborhood is bounded. Intuitively, if many nodes are located close to each other, many of them must be within mutual transmission range. As the model only restricts the number of independent nodes in each neighborhood, it is, therefore, a generalization of the unit disk graph (UDG), the quasi unit disk graph (QUDG), and the unit ball graph (UBG) models, which are essential model of WSNs [35].

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Acknowledgements

This work was supported by the Open University of Israel’s Research Fund and ISF grant 724/15. The authors are grateful to Michael Elkin for very helpful suggestions. The authors would also like to thank the reviewers for all of their careful, constructive, and insightful comments.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, The Open University of Israel, Ra’anana, Israel

    Leonid Barenboim

  2. Department of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel

    Gal Oren

  3. Scientific Computing Center, Nuclear Research Center - Negev, Be’er-Sheva, Israel

    Gal Oren

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  1. Leonid Barenboim
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  2. Gal Oren
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Correspondence to Gal Oren.

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Preliminary results of this paper appeared in APOCS 2020, SOSA 2020, ICDCN 2021 conferences.

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Barenboim, L., Oren, G. Distributed backup placement. Distrib. Comput. 35, 455–473 (2022). https://doi.org/10.1007/s00446-022-00423-z

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