Bounds for self-stabilization in unidirectional networks

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

Stabilizing leader election in population protocols

In this paper we address the stabilizing leader election problem in the population protocols model augmented with oracles. Population protocols is a recent model of computation that captures the interactions of biological systems. In this model emergent global behavior is observed while anonymous finite-state agents(nodes) perform local peer interactions. Uniform self-stabilizing leader election is impossible in such systems without additional assumptions. Therefore, the classical model has been augmented with the eventual leader detector, Omega?, that eventually detects the presence or absence of a leader. In the augmented model several solutions for leader election in rings and complete networks have been proposed. In this work we extend the study to trees and arbitrary topologies. We propose deterministic and probabilistic solutions. All the proposed algorithms are memory optimal --- they need only one memory bit per agent. Additionally, we prove the necessity of the eventual leader detector even in environments helped by randomization

Self-stabilizing minimum-degree spanning tree within one from the optimal degree

International audienceWe propose a self-stabilizing algorithm for constructing a Minimum-Degree Spanning Tree (MDST) in undirected networks. Starting from an arbitrary state, our algorithm is guaranteed to converge to a legitimate state describing a spanning tree whose maximum node degree is at most ∆∗ + 1, where ∆∗ is the minimum possible maximum degree of a spanning tree of the network. To the best of our knowledge our algorithm is the first self stabilizing solution for the construction of a minimum-degree spanning tree in undirected graphs. The algorithm uses only local communications (nodes interact only with the neighbors at one hop distance). Moreover, the algorithm is designed to work in any asynchronous message passing network with reliable FIFO channels. Additionally, we use a fine grained atomicity model (i.e. the send/receive atomicity). The time complexity of our solution is O(mn2 log n) where m is the number of edges and n is the number of nodes. The memory complexity is O(δ log n) in the send-receive atomicity model (δ is the maximal degree of the network)

log(n)-approximation d'un arbre de Steiner auto-stabilisant et dynamique

National audienceCe travail est motivé entre autre, par le maintient distribué d'infrastructures optimisées pour la communication d'un groupe d'utilisateurs dispersé sur un réseau dynamique. Les domaines d'application typiques de telles structures sont les systèmes de publish/subscribe, bases de données distribuées, systèmes multicasts. Dans ce papier nous décrivons un algorithme distribué qui construit et maintient un arbre de Steiner approché connectant un groupe dynamique de membres dispersé sur un réseau dynamique. Le coût de la solution retournée par notre algorithme est au plus $\log |S|$ fois le coût de la solution optimale, $S$ étant le groupe de membres à interconnecter. Notre algorithme améliore les solutions existantes de plusieurs façons. Premièrement, il tolère le dynamisme des membres et du réseau, autrement dit les membres peuvent rejoindre ou quitter le groupe et les noeuds ou liens du réseau peuvent apparaître ou disparaître du réseau. Deuxièmement notre algorithme est auto-stabilisant, en d'autres termes il tolère les fautes transitoires. Enfin, notre algorithme est super-stabilisant, ce qui signifie que l'on garantie des propriétés sur la structure construite durant la convergence de l'algorithme et malgré le dynamisme du réseau

Sur le Coloriage Auto-stabilisant dans les Réseaux Unidirectionnels Anonymes

International audienceNous considérons des réseaux unidirectionnels anonymes. Nous démontrons que contrairement aux réseaux bidirectionnels, l'auto-stabilisation de tâches locales peut y être aussi difficile que l'auto-stabilisation de tâches globales. Pour ce faire, nous prenons comme exemple le problème du coloriage des nœuds d'un réseau. Plus précisément, nous démontrons que le coloriage auto-stabilisant est intrinsèquement aussi difficile à résoudre de manière déterministe qu'une tâche globale. Nous proposons ensuite une approche probabiliste pour retrouver une complexité locale

Stabilizing Flocking Via Leader Election in Robot Networks

International audienceFlocking is the ability of a group of robots to follow a leader or head whenever it moves in a plane (two dimensional Cartesian space). In this paper we propose and prove correct an architecture for a self-organizing and stabilizing flocking system. Contrary to the existing work on this topic our flocking architecture does not rely on the existence of a specific leader a priori known to every robot in the network. In our approach robots are uniform, start in an arbitrary configuration and the head of the group is elected via algorithmic tools.Our contribution is threefold. First, we propose novel probabilistic solutions for leader election in asynchronous settings under bounded schedulers. Additionally, we prove the impossibility of deterministic leader election when robots have no common coordinates and start in an arbitrary configuration. Secondly, we propose a collision free deterministic algorithm for circle formation designed for asynchronous networks. Thirdly, we propose a deterministic flocking algorithm totally independent of the existence of an a priori known leader. The proposed algorithm also works in asynchronous networks