Toward More Localized Local Algorithms: Removing Assumptions Concerning Global Knowledge
Amos Korman · Jean-Sébastien Sereni · Laurent Viennot
Received: date / Accepted: date
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 [6], by Kuhn [22], and by Panconesi and Srinivasan [34], as well as the $O(\Delta^2)$-coloring algorithm by Linial [28]. 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.
Keywords distributed algorithm · global knowledge · parameters · MIS · coloring · maximal matching
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.
Amos Korman CNRS and University Paris Diderot LIAFA Case 7014 Université Paris Diderot – Paris 7 F-75205 Paris Cedex 13, France. Tel.: +33-1-57-27-92-56 Fax: +33-1-57-27-94-09 E-mail: Amos.Korman@liafa.jussieu.fr
Jean-Sébastien Sereni CNRS (LIAFA, Université Denis Diderot), Paris, France and Department of Applied Mathematics (KAM), Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic E-mail: sereni@kam.mff.cuni.cz
Laurent Viennot INRIA and University Paris Diderot LIAFA Case 7014 Université Paris Diderot – Paris 7 F-75205 Paris Cedex 13, France. E-mail: Laurent.Viennot@inria.fr
1 Introduction
1.1 Background and Motivation
Distributed computing concerns environments in which many processors, located at different sites, must collaborate in order to achieve some global task. One of the main themes in distributed network algorithms concerns the question of how to cope with locality constraints, that is, the lack of knowledge about the global structure of the network (cf., [35]). On the one hand, information about the global structure may not always be accessible to individual processors and the cost of computing it from scratch may overshadow the cost of the algorithm using it. On the other hand, global knowledge is not always essential, and many seemingly global tasks can be efficiently achieved by letting processors