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Toward more localized local algorithms: removing assumptions concerning global knowledge

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Abstract

Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and \((\Delta +1)\)-coloring algorithms by Barenboim and Elkin (Distrib Comput 22(5–6):363–379, 2010), by Kuhn (2009), and by Panconesi and Srinivasan (J Algorithms 20(2):356–374, 1996), as well as the \(O\mathopen {}(\Delta ^2)\)-coloring algorithm by Linial (J Comput 21:193, 1992). Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, local algorithms generally use good estimations of one or more global parameters of the network, e.g., the maximum degree \(\Delta \) or the number of nodes \(n\). This paper provides a method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to many results concerning the coloring problem (In particular, it applies to all aforementioned algorithms). To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.

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Notes

  1. In fact, we could have used in the definition of \(F_{\text{ med}}\) any small constant instead of \(1/3\), but \(1/3\) is sufficiently good for our purposes as, anyway, this result will be combined with better results for \(a=o\mathopen {}(\sqrt{\log n})\), which shall be established later on, in Corollary 4.

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Acknowledgments

The authors thank Boaz Patt-Shamir and the anonymous referees for their careful reading and thoughtful suggestions. Their comments helped to considerably improve the presentation of the paper.

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Authors and Affiliations

  1. CNRS and Université Paris Diderot, LIAFA Case 7014, 75205,  Paris Cedex 13, France

    Amos Korman & Jean-Sébastien Sereni

  2. Department of Applied Mathematics (KAM), Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Jean-Sébastien Sereni

  3. INRIA and Université Paris Diderot, LIAFA Case 7014, 75205 , Paris Cedex 13, France

    Laurent Viennot

Authors
  1. Amos Korman
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  2. Jean-Sébastien Sereni
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  3. Laurent Viennot
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Correspondence to Amos Korman.

Additional information

Amos Korman is supported in part by a France-Israel cooperation grant (“Mutli-Computing” project) from the France Ministry of Science and Israel Ministry of Science, by the ANR projects ALADDIN and PROSE, and by the INRIA project GANG. Jean-Sébastien Sereni is partially supported by the French Agence Nationale de la Recherche under reference anr 10 jcjc 0204 01. Laurent Viennot is supported by the european STREP project EULER, and the INRIA project-team GANG.

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Korman, A., Sereni, JS. & Viennot, L. Toward more localized local algorithms: removing assumptions concerning global knowledge. Distrib. Comput. 26, 289–308 (2013). https://doi.org/10.1007/s00446-012-0174-8

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