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A Note on the Parallel Runtime of Self-Stabilizing Graph Linearization

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Abstract

Topological self-stabilization is an important concept to build robust open distributed systems (such as peer-to-peer systems) where nodes can organize themselves into meaningful network topologies. The goal is to devise distributed algorithms where nodes forward, insert, and delete links to neighboring nodes, and that converge quickly to such a desirable topology, independently of the initial network configuration. This article proposes a new model to study the parallel convergence time. Our model sheds light on the achievable parallelism by avoiding bottlenecks of existing models that can yield a distorted picture. As a case study, we consider local graph linearization—i.e., how to build a sorted list of the nodes of a connected graph in a distributed and self-stabilizing manner. In order to study the main structure and properties of our model, we propose two variants of a most simple local linearization algorithm. For each of these variants, we present analyses of the worst-case and best-case parallel time complexities, as well as the performance under a greedy selection of the actions to be executed. It turns out that the analysis is non-trivial despite the simple setting, and to complement our formal insights we report on our experiments which indicate that the runtimes may be better in the average case.

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Notes

  1. In fact, most likely no such local mechanism exists for implementing the worst-case and best-case schedulers, while we believe that local distributed implementations that closely approximate—within a constant factor of the parallel complexity—the randomized and greedy schedulers presented here would not be hard to devise.

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Author notes

  1. Riko Jacob

    Present address: Eidgenössische Technische Hochschule (ETH) Zürich, Universitätstr. 6, 8092, Zürich, Switzerland

Authors and Affiliations

  1. Department of Computer Science, Institut für Informatik, Technische Universität München, Boltzmannstr. 3, 85748, Garching, Germany

    Dominik Gall, Riko Jacob & Hanjo Täubig

  2. Department of Computer Science and Engineering, Arizona State University, Box 878809, Tempe, AZ, 85287-8809, USA

    Andrea Richa

  3. Department of Computer Science, Institut für Informatik, Universität Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Christian Scheideler

  4. Telekom Innovation Labs & Technische Universität Berlin, Ernst-Reuter-Platz 7, 10587, Berlin, Germany

    Stefan Schmid

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  1. Dominik Gall
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  3. Andrea Richa
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  4. Christian Scheideler
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  5. Stefan Schmid
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  6. Hanjo Täubig
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Corresponding author

Correspondence to Stefan Schmid.

Additional information

A preliminary version of this work without all proofs and simulations has been presented at the 9th Latin American Theoretical Informatics Symposium (LATIN) 2010. This work is partly supported by the National Science Foundation (NSF grant CCF-0830704) and by the German Research Foundation (DFG project SCHE 1592/1-1 and SFB 901: On-the-Fly Computing)

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Gall, D., Jacob, R., Richa, A. et al. A Note on the Parallel Runtime of Self-Stabilizing Graph Linearization. Theory Comput Syst 55, 110–135 (2014). https://doi.org/10.1007/s00224-013-9504-x

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