# Bounds for self-stabilization in unidirectional networks

3 Regal - Large-Scale Distributed Systems and Applications
LIP6 - Laboratoire d'Informatique de Paris 6, Inria Paris-Rocquencourt
4 GRAND-LARGE - Global parallel and distributed computing
LRI - Laboratoire de Recherche en Informatique, LIFL - Laboratoire d'Informatique Fondamentale de Lille, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR8623
Abstract : A distributed algorithm is self-stabilizing if after faults and attacks hit the system and place it in some arbitrary global state, the systems recovers from this catastrophic situation without external intervention in finite time. Unidirectional networks preclude many common techniques in self-stabilization from being used, such as preserving local predicates. In this paper, we investigate the intrinsic complexity of achieving self-stabilization in unidirectional networks, and focus on the classical vertex coloring problem. When deterministic solutions are considered, we prove a lower bound of $n$ states per process (where $n$ is the network size) and a recovery time of at least $n(n-1)/2$ actions in total. We present a deterministic algorithm with matching upper bounds that performs in arbitrary graphs. When probabilistic solutions are considered, we observe that at least $\Delta + 1$ states per process and a recovery time of $\Omega(n)$ actions in total are required (where $\Delta$ denotes the maximal degree of the underlying simple undirected graph). We present a probabilistically self-stabilizing algorithm that uses $\mathtt{k}$ states per process, where $\mathtt{k}$ is a parameter of the algorithm. When $\mathtt{k}=\Delta+1$, the algorithm recovers in expected $O(\Delta n)$ actions. When $\mathtt{k}$ may grow arbitrarily, the algorithm recovers in expected $O(n)$ actions in total. Thus, our algorithm can be made optimal with respect to space or time complexity.
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Cited literature [21 references]

https://hal.inria.fr/inria-00277661
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Submitted on : Tuesday, May 13, 2008 - 10:01:34 AM
Last modification on : Sunday, June 26, 2022 - 11:48:14 AM
Long-term archiving on: : Tuesday, September 21, 2010 - 4:11:22 PM

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RR-6524.pdf
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### Identifiers

• HAL Id : inria-00277661, version 2
• ARXIV : 0805.0851

### Citation

Samuel Bernard, Stéphane Devismes, Maria Gradinariu Potop-Butucaru, Sébastien Tixeuil. Bounds for self-stabilization in unidirectional networks. [Research Report] RR-6524, INRIA. 2008, pp.24. ⟨inria-00277661v2⟩

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