Algorithme Auto-Stabilisant Compact d'Election pour les Graphes Arbitraires

International audienceNous présentons le premier algorithme auto-stabilisant d'´ election pour les réseaux de topologie arbitraire dont la complexité en espace est de O(max{log ∆, log log n}) bits par noeud, o` u n est la taille du réseau et ∆ son degré. Cette complexité en espace est sous-logarithmique en n, tant que ∆ = n o(1)

Vers une structuration auto-stabilisante des réseaux Ad Hoc

International audienceIn this paper, we present a self-stabilizing asynchronous distributed clustering algorithm that builds non-overlapping k-hops clusters. Our approach does not require any initialization. It is based only on information from neighboring nodes with periodic messages exchange. Starting from an arbitrary configuration, the network converges to a stable state after a finite number of steps. Firstly, we prove that the stabilization is reached after at most n+2 transitions and requires (u+1)* log(2n+k+3) bits per node, whereΔu represents node's degree, n is the number of network nodes and k represents the maximum hops number. Secondly, using OMNet++ simulator, we performed an evaluation of our proposed algorithm.Dans cet article, nous proposons un algorithme de structuration auto-stabilisant, distribuéet asynchrone qui construit des clusters de diamètre au plus 2k. Notre approche ne nécessite aucuneinitialisation. Elle se fonde uniquement sur l’information provenant des noeuds voisins à l’aided’échanges de messages. Partant d’une configuration quelconque, le réseau converge vers un étatstable après un nombre fini d’étapes. Nous montrons par preuve formelle que pour un réseau de nnoeuds, la stabilisation est atteinte en au plus n + 2 transitions. De plus, l’algorithme nécessite uneoccupation mémoire de (u + 1) log(2n + k + 3) bits pour chaque noeud u où u représente ledegré (nombre de voisins) de u et k la distance maximale dans les clusters. Afin de consolider lesrésultats théoriques obtenus, nous avons effectué une campagne de simulation sous OMNeT++ pourévaluer la performance de notre solution

The Lattice structure of Chip Firing Games and Related Models

In this paper, we study a famous discrete dynamical system, the Chip Firing Game, used as a model in physics, economics and computer science. We use order theory and show that the set of reachable states (i.e. the configuration space) of such a system started in any configuration is a lattice, which implies strong structural properties. The lattice structure of the configuration space of a dynamical system is of great interest since it implies convergence (and more) if the configuration space is finite. If it is infinite, this property implies another kind of convergence: all the configurations reachable from two given configurations are reachable from their infimum. In other words, there is a unique first configuration which is reachable from two given configurations. Moreover, the Chip Firing Game is a very general model, and we show how known models can be encoded as Chip Firing Games, and how some results about them can be deduced from this paper. Finally, we define a new model, which is a generalization of the Chip Firing Game, and about which many interesting questions arise.Comment: See http://www.liafa.jussieu.fr/~latap

Algorithmique distribuée, calculs locaux et homomorphismes de graphes

Dans cette thèse, on étudie ce qui est calculable dans différents modèles d’algorithmique distribuée. Les modèles considérés correspondent à différents niveaux d’abstraction et à différents niveaux de synchronisation entre les processus d’un système distribué. On s’intéresse en particulier au problèmes de l’élection et du nommage dans ces différents modèles. Pour chaque modèle, on caractérise les systèmes distribués dans lesquels on peut résoudre ces problèmes et on étudie la complexité des problèmes de décision correspondants. Nos caractérisations utilisent des homomorphismes de graphes qui préservent certaines propriétés locales. Nos preuves sont constructives : quand on peut résoudre l’élection (ou le nommage) dans un réseau, on présente un algorithme d’élection (ou de nommage) pour ce réseau. Ces problèmes permettent de mettre en évidence les différences entre les puissances de calculs des différents modèles considérés. De plus, l’étude de ces problèmes permet de mettre à jour les bons outils qui permettent d’étudier ce qui est calculable de manière distribuée dans les différents modèles.In this thesis, we consider different models of distributed computations. These models correspond to different levels of abstraction and they encode different levels of synchronization between processes in a distributed system. In these different models, we particularly focus on two classical problems in distributed computing : election and naming. For each model, we present a characterization of distributed systems where these problems can be solved and we study the complexity of the corresponding decision problems. Our characterizations are expressed in terms of graph homomorphisms that preserve some local properties. Our proofs are constructive : when a network admits an election (or a naming) algorithm, we present such an algorithm for this network. These problems enable to highlight the differences between the computation powers of the different models we consider. Moreover, studying these problems enable to introduce some combinatorial and algorithmic tools that can be used to study what can be computed in a distributed way in these different models

Optimal Space Lower Bound for Deterministic Self-Stabilizing Leader Election Algorithms

International audienceGiven a boolean predicate Π on labeled networks (e.g., proper coloring, leader election, etc.), a self-stabilizing algorithm for Π is a distributed algorithm that can start from any initial configuration of the network (i.e., every node has an arbitrary value assigned to each of its variables), and eventually converge to a configuration satisfying Π. It is known that leader election does not have a deterministic self-stabilizing algorithm using a constant-size register at each node, i.e., for some networks, some of their nodes must have registers whose sizes grow with the size n of the networks. On the other hand, it is also known that leader election can be solved by a deterministic self-stabilizing algorithm using registers of O(log log n) bits per node in any n-node bounded-degree network. We show that this latter space complexity is optimal. Specifically, we prove that every deterministic self-stabilizing algorithm solving leader election must use Ω(log log n)-bit per node registers in some n-node networks. In addition, we show that our lower bounds go beyond leader election, and apply to all problems that cannot be solved by anonymous algorithms

Self-stabilizing leader election in optimal space under an arbitrary scheduler

AbstractA silent self-stabilizing asynchronous distributed algorithm, SSLE, is given for the leader election problem in a connected unoriented (bidirectional) network with unique IDs. SSLE also constructs a BFS tree on the network rooted at that leader. SSLE uses O(logn) space per process and stabilizes in O(n) rounds, against the unfair daemon, where n is the number of processes in the network