Beeping a Deterministic Time-Optimal Leader Election

The beeping model is an extremely restrictive broadcast communication model that relies only on carrier sensing. In this model, we solve the leader election problem with an asymptotically optimal round complexity of O(D + log n), for a network of unknown size n and unknown diameter D (but with unique identifiers). Contrary to the best previously known algorithms in the same setting, the proposed one is deterministic. The techniques we introduce give a new insight as to how local constraints on the exchangeable messages can result in efficient algorithms, when dealing with the beeping model. Using this deterministic leader election algorithm, we obtain a randomized leader election algorithm for anonymous networks with an asymptotically optimal round complexity of O(D + log n) w.h.p. In previous works this complexity was obtained in expectation only. Moreover, using deterministic leader election, we obtain efficient algorithms for symmetry-breaking and communication procedures: O(log n) time MIS and 5-coloring for tree networks (which is time-optimal), as well as k-source multi-broadcast for general graphs in O(min(k,log n) * D + k log{(n M)/k}) rounds (for messages in {1,..., M}). This latter result improves on previous solutions when the number of sources k is sublogarithmic (k = o(log n))

Brief Announcement: Distributed Exclusive and Perpetual Tree Searching

International audienceWe tackle a practical version of the well known {\it graph searching} problem, where a team of robots aims at capturing an intruder in a graph. The robots and the intruder move along the edges of the graph. The intruder is invisible, arbitrary fast, and omniscient. It is caught whenever it stands on a node occupied by a robot, and cannot escape to a neighboring node. We study graph searching in the CORDA model of mobile computing: robots are asynchronous, and they perform cycles of {\it Look-Compute-Move} actions. Moreover, motivated by physical constraints, we consider the \emph{exclusive} property, stating that no two or more robots can occupy the same node at the same time. In addition, we assume that the network and the robots are anonymous. Finally, robots are \emph{oblivious}, i.e., each robot performs its move actions based only on its current ''vision'' of the positions of the other robots. Our objective is to characterize, for a graph $G$, the set of integers $k$ such that graph searching can be achieved by a team of $k$ robots starting from \emph{any} $k$ distinct nodes in $G$. Our main result consists in a full characterization of this set, for any asymmetric tree. Towards providing a characterization for all trees, including trees with non-trivial automorphisms, we have also provides a set of positive and negative results, including a full characterization for any line. All our positive results are based on the design of algorithms enabling \emph{perpetual} graph searching to be achieved with the desired number of robots

Nettoyage perpétuel de réseaux

International audienceDans le cadre du nettoyage de graphes contaminés ( graph searching), des agents mobiles se déplacent successivement le long des arêtes du graphe afin de les nettoyer. Le but général est le nettoyage en utilisant le moins d'agents possible. Nous plaçons notre étude dans le modèle de calcul distribué CORDA minimaliste. Ce modèle est muni d'hypothèses très faibles : les nœuds du réseau et les agents sont anonymes, n'ont pas de mémoire du passé ni sens commun de l'orientation et agissent par cycles Voir-Calculer-Agir de manière asynchrone. Un intérêt de ce modèle vient du fait que si le nettoyage peut être fait à partir de positions arbitraires des agents (par exemple, après pannes ou recontamination), l'absence de mémoire implique un nettoyage perpétuel et donc fournit une première approche de nettoyage de graphe tolérant aux pannes. Les contraintes dues au modèle CORDA minimaliste nous amènent à définir une nouvelle variante de nettoyage de graphes - le nettoyage sans collision, autrement dit, plusieurs agents ne peuvent occuper simultanément un même sommet. Nous montrons que, dans un contexte centralisé, cette variante ne satisfait pas certaines propriétés classiques de nettoyage comme par exemple la monotonie. Nous montrons qu'interdire les ''collisions'' peut augmenter le nombre d'agents nécessaires d'un facteur au plus $\Delta$ le degré maximum du graphe et nous illustrons cette borne. De plus, nous caractérisons complètement le nettoyage sans collision dans les arbres. Dans le contexte distribué, la question qui se pose est la suivante. Existe-t-il un algorithme qui, étant donné un ensemble d'agents mobiles arbitrairement répartis sur des sommets distincts d'un réseau, permet aux agents de nettoyer perpétuellement le graphe ? Dans le cas des chemins, nous montrons que la réponse est négative si le nombre d'agents est pair dans un chemin d'ordre impair, ou si il y a au plus deux agents dans un chemin d'ordre au moins $3$. Nous proposons un algorithme qui nettoie les chemins dans tous les cas restants, ainsi qu'un algorithme pour nettoyer les arbres lorsqu'un nombre suffisant d'agents est disponible initialement

The Benefits of Entropy in Population Protocols

A distributed computing system can be viewed as the result of the interplay between a distributed algorithm specifying the effects of a local event (e.g. reception of a message), and an adversary choosing the interleaving (schedule) of these events in the execution. In the context of large networks of mobile pairwise interacting agents (population protocols), the adversary models the mobility of the agents by choosing the successive pairs of interacting agents. For some problems, assuming that the adversary selects the schedule according to some probability distribution greatly helps to devise (almost) correct solutions. But how much randomness is really necessary? To what extent does a problem admit implementations that are robust against a "not so random" schedule? This paper takes a first step in addressing this question by borrowing the concept of T-randomness, 0 <= T <= 1, from algorithmic information theory. Roughly speaking, the value T fixes the entropy rate of the considered schedules. For instance, the case T = 1 corresponds, in a specific sense, to schedules in which the pairs of interacting agents are chosen independently and uniformly (perfect randomness). The holy grail question can then be precisely stated as determining the optimal entropy rate to solve a given problem. We first show that perfect randomness is never required. Precisely, if a finite-state algorithm solves a problem with 1-randomness, then this algorithm still solves the same problem with T-randomness for some T < 1. Second, we illustrate how to compute bounds on the optimal entropy rate of a specific problem, namely the leader election problem

Non-deterministic Population Protocols (Extended Version)

In this paper we show that, in terms of generated output languages, non-deterministic \textit{population protocols} are strictly more powerful than deterministic ones. Analyzing the reason for this negative result, we propose two slightly enhanced models, in which non-deterministic population protocols can be \emph{exactly} simulated by deterministic ones. First, we consider a model in which interactions are not only between couples of agents, but also between triples and in which non-uniform initial states are allowed. We generalize this transformation and we prove a general property for a model with interactions between any number of agents. Second, we simulate any non-deterministic population protocol by a deterministic one in a model where a \emph{configuration} can have an \textit{empty output}. Non-deterministic and deterministic population protocols are then compared in terms of inclusion of their output languages, that is, in terms of solvability of problems. Two transformations, which realize this inclusion, are presented. The first one uses (again) the natural model with interactions of triples, but does not need non-uniform initial states. As before, this result is generalized for the natural model with interactions between any number of agents. The second transformation is a parameterized one with parameters depending on the transition graph of the considered non-deterministic protocol and on the population. Note that the transformations in the paper apply to a whole class of non-deterministic population protocols (for a proposed model), in contrast with the transformations proposed in previous works, which apply only to a specific sub-class of protocols (satisfying a so called ''elasticity'' condition)

Self-stabilizing Mutual Exclusion and Group Mutual Exclusion for Population Protocols with Covering

International audienceThis paper presents and proves correct two self-stabilizing deterministic algorithms solving the mutual exclusion and the group mutual exclusion problems in the model of population protocols with covering. In this variant of the population protocol model, a local fairness is used and bounded state anonymous mobile agents interact in pairs according to constraints expressed in terms of their cover times. The cover time is an indicator of the "time" for an agent to communicate with all the other agents. This indicator is expressed in the number of the pairwise communications (events) and is unknown to agents. In the model, we also assume the existence of a particular agent, the base station. In contrast with the other agents, it has a memory size proportional to the number of the agents. We prove that without this kind of assumption, the mutual exclusion problem has no solution. The algorithms in the paper use a phase clock tool. This is a synchronization tool that was recently proposed in the model we use. For our needs, we extend the functionality of this tool to support also phases with unbounded (but finite) duration. This extension seems to be useful also in the future works

Self-stabilizing Synchronization in Mobile Sensor Networks with Covering

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Time Optimal Asynchronous Self-stabilizing Spanning Tree

Abstract. This paper presents an improved and time-optimal selfstabilizing algorithm for a major task in distributed computing- a rooted spanning tree construction. Our solution is decentralized (“truly distributed”), uses a bounded memory and is not based on the assumption that either n (the number of nodes), or diam (the actual diameter of the network), or an existence of cycles in the network are known. The algorithm assumes asynchronous and reliable FIFO message passing and unique identifiers, and works in dynamic networks and for any network topology. One of the previous time-optimal algorithms for this task was designed for a model with coarse-grained atomic operations and can be shown not to work properly for the totally asynchronous model (with just “read” or “receive ” atomicity, and “write ” or “send ” atomicity). We revised the algorithm and proved it for a more realistic model of totally asynchronous networks. The state in the presented algorithm does not stabilize until long after the required output does. For such an algorithm, an increased asynchrony poses much increased hardness in the proof.