Compact Deterministic Self-Stabilizing Leader Election: The Exponential Advantage of Being Talkative

This paper focuses on compact deterministic self-stabilizing solutions for the leader election problem. When the protocol is required to be \emph{silent} (i.e., when communication content remains fixed from some point in time during any execution), there exists a lower bound of Omega(\log n) bits of memory per node participating to the leader election (where n denotes the number of nodes in the system). This lower bound holds even in rings. We present a new deterministic (non-silent) self-stabilizing protocol for n-node rings that uses only O(\log\log n) memory bits per node, and stabilizes in O(n\log^2 n) rounds. Our protocol has several attractive features that make it suitable for practical purposes. First, the communication model fits with the model used by existing compilers for real networks. Second, the size of the ring (or any upper bound on this size) needs not to be known by any node. Third, the node identifiers can be of various sizes. Finally, no synchrony assumption, besides a weakly fair scheduler, is assumed. Therefore, our result shows that, perhaps surprisingly, trading silence for exponential improvement in term of memory space does not come at a high cost regarding stabilization time or minimal assumptions

On the Limits and Practice of Automatically Designing Self-Stabilization

A protocol is said to be self-stabilizing when the distributed system executing it is guaranteed to recover from any fault that does not cause permanent damage. Designing such protocols is hard since they must recover from all possible states, therefore we investigate how feasible it is to synthesize them automatically. We show that synthesizing stabilization on a fixed topology is NP-complete in the number of system states. When a solution is found, we further show that verifying its correctness on a general topology (with any number of processes) is undecidable, even for very simple unidirectional rings. Despite these negative results, we develop an algorithm to synthesize a self-stabilizing protocol given its desired topology, legitimate states, and behavior. By analogy to shadow puppetry, where a puppeteer may design a complex puppet to cast a desired shadow, a protocol may need to be designed in a complex way that does not even resemble its specification. Our shadow/puppet synthesis algorithm addresses this concern and, using a complete backtracking search, has automatically designed 4 new self-stabilizing protocols with minimal process space requirements: 2-state maximal matching on bidirectional rings, 5-state token passing on unidirectional rings, 3-state token passing on bidirectional chains, and 4-state orientation on daisy chains

Distributed stabilizing data structures

Distributed algorithms aim to achieve better performance than sequential algorithms in terms of time complexity (or asymptotic time complexity) while keeping or lowering the memory requirement (space complexity) in a node. (In sequential algorithms, the memory requirement is the memory requirement of the algorithm itself.); Self-stabilizing distributed algorithms aim to achieve a comparable performance to non-stabilizing distributed algorithms when transient faults or arbitrary initialization cause the system to enter a state where a non-stabilizing algorithm cannot continue to properly perform its task; Transient faults can affect an existing data structure and alter its data content. As a result, the data structure may lose its properties, and the operations defined over the data structure will have unpredictable and undesirable results, making the data structure unusable; We present several self or snap-stabilizing algorithms for particular data structures; We propose an optimal self-stabilizing distributed algorithm for simultaneously activating non-adjacent processes on an oriented chain (Algorithm SSDS ). We use Algorithm SSDS to accomplish two tasks: local mutual exclusion and line sorting. We propose two uniform, self-stabilizing, deterministic protocols on oriented chains: a time and space optimal solution to the local mutual exclusion problem (Algorithm LMEC ), and a space and (asymptotic) time optimal solution to the distributed sorting problem (Algorithm SORTc ); We extend Algorithm SSDS to an asynchronous oriented ring with a distinguished node with some minor modifications, and we obtain general self-stabilization for simultaneously activated non-adjacent processes in an oriented ring with a distinguished process (Algorithm SSDSR ). We use Algorithm SSDSR to accomplish two tasks: local resource allocation and ring sorting. We propose two uniform, self-stabilizing, deterministic protocols on oriented rings: a time and space optimal solution to the local resource allocation problem (Algorithm LRAR ), and a space and (asymptotic) time optimal solution to the distributed sorting problem (Algorithm SORTr ); We extend Algorithm SSDS to an asynchronous rooted tree, and we obtain general self-stabilization for simultaneously activated non-adjacent processes in a rooted tree (Algorithm SSDST ). We then give two applications of Algorithm SSDST : a time and space optimal solution to the local mutual exclusion problem (Algorithm LMET ) and a space and (asymptotically) time optimal solution to the min heap problem (Algorithm HEAP ); In proving the time complexity of sorting, we introduce the notion of pseudo-time, similar to logical time introduced by Lamport; We present the first snap-stabilizing distributed binary search tree (BST) algorithm. The proposed algorithm uses a heap algorithm (Algorithm Heap) as a preprocessing step. This is also the first snap-stabilizing distributed solution to the heap problem

Analyse de quelques algorithmes probabilistes à délais aléatoires

Dans la première partie de cette étude, nous proposons et analysons des algorithmes probabilistes d’élection uniforme dans des graphes de types arbres, les k-arbres et les polyominoïdes. Ces algorithmes utilisent des durées de vie aléatoires associées aux sommets découverts (sommets feuilles ou simpliciaux). Ces durées sont des variables aléatoires indépendantes et sont localement engendrées au fur et à mesure que les sommets sont découverts. Dans la seconde partie, nous analysons un algorithme probabiliste de synchronisation pour le problème de rendez-vous avec agendas dynamiques. L’objectif est de trouver un couplage maximal dans un graphe donné. Ensuite, nous proposons et étudions un modèle de diffusion à délai aléatoire pour la transmission d’un message dans un réseau. Finalement, dans la dernière partie, nous exposons les outils utilisés pour implémenter la simulation des algorithmes distribués.In the first part of this study, we propose and analyze a probabilistic algorithms of uniform election in graphs of structures of the trees type, k-trees and polyominoids. These algorithms use random delay associated to discovered vertices (leaf vertices or simplicial vertices). These delays are independent random variables and are locally generated as and when the vertices are discovered. In the second part, we analyze a probabilistic algorithm of synchronization for the problem of rendezvous with dynamic agendas. The goal is to find a maximal matching in a given graph. Then, we propose and study a model of diffusion with random delay for the transmission of a message in a network. Finally, in the last part, we expose the tools used to implement the simulation of the distributed algorithms