HAL : hal-00307375, version 1
 Foundations of System Specification and Computation Structures (FoSSaCS) (EATCS best paper award), Spain (2002)
 A characterization of families of graphs in which election is possible
 (2002)
 In this paper, using some techniques developed for the termination detection problem, we characterize which knowledge is necessary and sufficient to have an election algorithm, or equivalently, what is the general condition for a class of graphs to admit an election algorithm. More precisely we prove the following theorem: it There is an election algorithm for a family I of graphs if and only if graphs of I are minimal for the covering relation and every graph G of I has quasi-coverings of bounded radius in I. Sufficient conditions given below are just special cases of criteria of the theorem. We explain new parts in this theorem. It is well known (see above) that the existence of an election algorithm needs graphs minimal for the covering relation. We prove in this paper that if a graph is minimal for the covering relation and admits quasi-coverings of arbitrary large size in the family there is no election algorithm. This part can be illustrated by the family of prime rings. Indeed, prime rings are minimal for the covering relation nevertheless there is no election algorithm for this family: without the knowledge of the size, a ring admits quasi-covering prime rings of arbitrary large size. These two results prove one direction of the theorem . To prove the converse: we extend the Mazurkiewicz algorithm to labelled graphs; we prove that the Mazurkiewicz algorithm applied in a graph G enables the reconstruction, on each node of G, of a graph K such that G is a quasi-covering of K; and when the computation is terminated G is a covering of K; we use an extension of an algorithm by Szymanski, Shi and Prywes which enables the distributed detection of stable properties in a graph; we prove that the bounded size of quasi-coverings of a given graph enables to each node v to detect the termination of the Mazurkiewicz algorithm and finally each node can decide if it has obtained the maximum number among numbers computed by the Mazurkiewicz algorithm.
 1 : Laboratoire d'informatique Fondamentale de Marseille (LIF) CNRS : UMR6166 – Université de la Méditerranée - Aix-Marseille II – Université de Provence - Aix-Marseille I 2 : Laboratoire Bordelais de Recherche en Informatique (LaBRI) CNRS : UMR5800 – Université Sciences et Technologies - Bordeaux I – École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB) – Université Victor Segalen - Bordeaux II
 Domaine : Informatique/Autre
 hal-00307375, version 1 http://hal.archives-ouvertes.fr/hal-00307375 oai:hal.archives-ouvertes.fr:hal-00307375 Contributeur : Yves Métivier <> Soumis le : Mardi 29 Juillet 2008, 11:11:20 Dernière modification le : Mercredi 29 Octobre 2008, 16:36:28