Termination Detection of Local Computations

Contrary to the sequential world, the processes involved in a distributed system do not necessarily know when a computation is globally finished. This paper investigates the problem of the detection of the termination of local computations. We define four types of termination detection: no detection, detection of the local termination, detection by a distributed observer, detection of the global termination. We give a complete characterisation (except in the local termination detection case where a partial one is given) for each of this termination detection and show that they define a strict hierarchy. These results emphasise the difference between computability of a distributed task and termination detection. Furthermore, these characterisations encompass all standard criteria that are usually formulated : topological restriction (tree, rings, or triangu- lated networks ...), topological knowledge (size, diameter ...), and local knowledge to distinguish nodes (identities, sense of direction). These results are now presented as corollaries of generalising theorems. As a very special and important case, the techniques are also applied to the election problem. Though given in the model of local computations, these results can give qualitative insight for similar results in other standard models. The necessary conditions involve graphs covering and quasi-covering; the sufficient conditions (constructive local computations) are based upon an enumeration algorithm of Mazurkiewicz and a stable properties detection algorithm of Szymanski, Shi and Prywes

Election and Local Computations on Edges

International audienceThe point of departure and the motivation for this paper are the results of Angluin [1] which has introduced a tool to analyze the election algorithm: the coverings, Yamashita and Kameda [21] and Mazurkiewicz [15] which have obtained characterizations of graphs in which election is possible under two different models of distributed computations. Our aim is twofold. First it is to obtain characterizations of graphs in which election is possible under intermediate models between the models of Yamashita-Kameda and of Mazurkiewicz. Our second aim is to understand the implications of the models for the borderline between positive and negative results for distributed computations. In this work, characterizations are obtained under three different models

A Bridge Between the Asynchronous Message Passing Model and Local Computations in Graphs

International audienceA distributed system is a collection of processes that can interact. Three major process interaction models in distributed systems have principally been considered: - the message passing model, - the shared memory model, - the local computation model. In each model the processes are represented by vertices of a graph and the interactions are represented by edges. In the message passing model and the shared memory model, processes interact by communication primitives: messages can be sent along edges or atomic read/write operations can be performed on registers associated with edges. In the local computation model interactions are defined by labelled graph rewriting rules; supports of rules are edges or stars. These models (and their sub-models) reflect different system architectures, different levels of synchronization and different levels of abstraction. Understanding the power of various models, the role of structural network properties and the role of the initial knowledge enhances our understanding of basic distributed algorithms. This is done with some typical problems in distributed computing: election, naming, spanning tree construction, termination detection, network topology recognition, consensus, mutual exclusion. Furthermore, solutions to these problems constitute primitive building blocks for many other distributed algorithms. A survey may be found in [FR03], this survey presents some links with several parameters of the models including synchrony, communication media and randomization. An important goal in the study of these models is to understand some relationships between them. This paper is a contribution to this goal; more precisely we establish a bridge between tools and results presented in [YK96] for the message passing model and tools and results presented in [Ang80, BCG+96, Maz97, CM04, CMZ04, Cha05] for the local computation mode

Election, Naming and Cellular Edge Local Computations

International audienceWe examine the power and limitations of the weakest vertex relabelling system which allows to change a label of a vertex in function of its own label and of the label of one of its neighbours. We characterise the graphs for which two important distributed algorithmic problems are solvable in this model: naming and election

Enumeration and Leader Election in Partially Anonymous and Multi-hop Broadcast Networks

We address the enumeration and the leader election problems over partially anonymous and multi-hop broadcast networks. We consider an asynchronous communication model where each process broadcasts a message and all its neighbours receive this message after arbitrary and unpredictable time. In this paper, we present necessary conditions that must be satisfied by any graph to solve these problems and we show that these conditions are sufficient by providing an enumeration algorithm on the one hand and a leader election algorithm on the other hand. From the complexity viewpoint, our algorithms offer a polynomial complexity (memory, number of messages and size of messages)

Snapshots et Détection de Propriétés Stables dans les Systèmes Distribués Anonymes

International audienceNous étudions les problèmes du calcul d'état global (ou snapshot) et, plus généralement, de la détection de propriétés stables dans les systèmes totalement distribués et anonymes. Nous considérons le modèle classique à passage de messages dans lequel, pour une étape de calcul, chaque élément du système peut changer son état, envoyer ou recevoir un message à travers des liens de communication. La plupart des algorithmes existants pour résoudre le problème du calcul d'état global supposent que les éléments du système ont des identifiants uniques ou qu'il existe un unique noeud initiateur. Ce travail concerne le calcul d'état global dans les systèmes anonymes et plus généralement quelles sont les propriétés stables d'un système distribué qui peuvent être détectées anonymement par l'utilisation de snapshots locaux tout en autorisant des initiateurs multiples et en ne connaissant qu'une borne supérieure sur le diamètre du réseau

Higher-dimensional normalisation strategies for acyclicity

We introduce acyclic polygraphs, a notion of complete categorical cellular model for (small) categories, containing generators, relations and higher-dimensional globular syzygies. We give a rewriting method to construct explicit acyclic polygraphs from convergent presentations. For that, we introduce higher-dimensional normalisation strategies, defined as homotopically coherent ways to relate each cell of a polygraph to its normal form, then we prove that acyclicity is equivalent to the existence of a normalisation strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical finiteness condition for higher categories which extends Squier's finite derivation type for monoids. We relate this homotopical property to a new homological finiteness condition that we introduce here.Comment: Final versio

Une condition suffisante de reconnaissabilité dans un monoïde partiellement commutatif

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