A Framework for Certified Self-Stabilization

We propose a general framework to build certified proofs of distributed self-stabilizing algorithms with the proof assistant Coq. We first define in Coq the locally shared memory model with composite atomicity, the most commonly used model in the self-stabilizing area. We then validate our framework by certifying a non trivial part of an existing silent self-stabilizing algorithm which builds a $k$-hop dominating set of the network. We also certified a quantitative property related to the output of this algorithm. Precisely, we show that the computed $k$-hop dominating set contains at most $\lfloor \frac{n-1}{k+1} \rfloor + 1$ nodes, where $n$ is the number of nodes in the network. To obtain these results, we also developed a library which contains general tools related to potential functions and cardinality of sets

A Self-Stabilizing K-Clustering Algorithm Using an Arbitrary Metric (Revised Version of RR2008-31)

32 pagesMobile ad hoc networks as well as grid platforms are distributed, changing, and error prone environments. Communication costs within such infrastructure can be improved, or at least bounded, by using k-clustering. A k-clustering of a graph, is a partition of the nodes into disjoint sets, called clusters, in which every node is distance at most k from a designated node in its cluster, called the clusterhead. A self-stabilizing asynchronous distributed algorithm is given for constructing a k-clustering of a connected network of processes with unique IDs and weighted edges. The algorithm is comparison-based, takes O(nk) time, and uses O(log n + log k) space per process, where n is the size of the network. This is the first distributed solution to the k-clustering problem on weighted graphs

Self-stabilizing k-clustering in mobile ad hoc networks

In this thesis, two silent self-stabilizing asynchronous distributed algorithms are given for constructing a k-clustering of a connected network of processes. These are the first self-stabilizing solutions to this problem. One algorithm, FLOOD, takes O( k) time and uses O(k log n) space per process, while the second algorithm, BFS-MIS-CLSTR, takes O(n) time and uses O(log n) space; where n is the size of the network. Processes have unique IDs, and there is no designated leader. BFS-MIS-CLSTR solves three problems; it elects a leader and constructs a BFS tree for the network, constructs a minimal independent set, and finally a k-clustering. Finding a minimal k-clustering is known to be NP -hard. If the network is a unit disk graph in a plane, BFS-MIS-CLSTR is within a factor of O(7.2552k) of choosing the minimal number of clusters; A lower bound is given, showing that any comparison-based algorithm for the k-clustering problem that takes o( diam) rounds has very bad worst case performance; Keywords: BFS tree construction, K-clustering, leader election, MIS construction, self-stabilization, unit disk graph

Algorithmes auto-stabilisants pour la construction de structures couvrantes réparties

This thesis deals with the self-stabilizing construction of spanning structures over a distributed system. Self-stabilization is a paradigm for fault-tolerance in distributed algorithms. It guarantees that the system eventually satisfies its specification after transient faults hit the system. Our model of distributed system assumes locally shared memories for communicating, unique identifiers for symmetry-breaking, and distributed daemon for execution scheduling, that is, the weakest proper daemon. More generally, we aim for the weakest possible assumptions, such as arbitrary topologies, in order to propose the most versatile constructions of distributed spanning structures. We present four original self-stabilizing algorithms achieving k-clustering, (f,g)-alliance construction, and ranking. For every of these problems, we prove the correctness of our solutions. Moreover, we analyze their time and space complexity using formal proofs and simulations. Finally, for the (f,g)-alliance problem, we consider the notion of safe convergence in addition to self-stabilization. It enforces the system to first quickly satisfy a specification that guarantees a minimum of conditions, and then to converge to a more stringent specification.Cette thèse s'intéresse à la construction auto-stabilisante de structures couvrantes dans un système réparti. L'auto-stabilisation est un paradigme pour la tolérance aux fautes dans les algorithmes répartis. Plus précisément, elle garantit que le système retrouve un comportement correct en temps fini après avoir été perturbé par des fautes transitoires. Notre modèle de système réparti se base sur des mémoires localement partagées pour la communication, des identifiants uniques pour briser les symétries et un ordonnanceur inéquitable, c'est-à-dire le plus faible des ordonnanceurs. Dans la mesure du possible, nous nous imposons d'utiliser les plus faibles hypothèses, afin d'obtenir les constructions les plus générales de structures couvrantes réparties. Nous présentons quatre algorithmes auto-stabilisants originaux pour le k-partitionnement, la construction d'une (f,g)-alliance et l'indexation. Pour chacun de ces problèmes, nous prouvons la correction de nos solutions. De plus, nous analysons leur complexité en temps et en espace à l'aide de preuves formelles et de simulations. Enfin, pour le problème de (f,g)-alliance, nous prenons en compte la notion de convergence sûre qui vient s'ajouter à celle d'auto-stabilisation. Elle garantit d'abord que le comportement du système assure rapidement un minimum de conditions, puis qu'il continue de converger jusqu'à se conformer à une spécification plus exigeante

Compact routing in fault-tolerant distributed systems

A compact routing algorithm is a routing algorithm which reduces the space complexity of all-pairs shortest path routing. Compact routing protocols in distributed systems have been studied extensively as an attractive alternative to the traditional method of all-pairs shortest path routing. The use of compact routing protocols have several advantages. Compact routing schemes are not only more memory-efficient, but provide faster routing table lookup, more efficient broadcast scheme, and allow for a more scalable network. These routing schemes still maintain optimal or near-optimal routing paths. However, most of the compact routing protocols are not fault-tolerant. This thesis will first report the recent developments in the compact routing research. Several new methods for compact routing in fault-tolerant distributed systems will be presented and analyzed. The most important feature of the algorithms presented in this thesis is that they are self-stabilizing. The self-stabilization paradigm has been shown to be the most unified and all-inclusive approach to the design of fault-tolerant system. Additionally, these algorithms will address and solve several problems left unsolved by previous works. Relabelable and non-relabelable networks will be considered for both specific and arbitrary topologies

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

Self-* distributed query region covering in sensor networks

Wireless distributed sensor networks are used to monitor a multitude of environments for both civil and military applications. Sensors may be deployed to unreachable or inhospitable areas. Thus, they cannot be replaced easily. However, due to various factors, sensors\u27 internal memory, or the sensors themselves, can become corrupted. Hence, there is a need for more robust sensor networks. Sensors are most commonly densely deployed, but keeping all sensors continually active is not energy efficient. Our aim is to select the minimum number of sensors which can entirely cover a particular monitored area, while remaining strongly connected. This concept is called a Minimum Connected Cover of a query region in a sensor network. In this research, we have designed two fully distributed, robust, self-* solutions to the minimum connected cover of query regions that can cope with both transient faults and sensor crashes. We considered the most general case in which every sensor has a different sensing and communication radius. We have also designed extended versions of the algorithms that use multi-hop information to obtain better results utilizing small atomicity (i.e., each sensor reads only one of its neighbors\u27 variables at a time, instead of reading all neighbors\u27 variables). With this, we have proven self-* (self-configuration, self-stabilization, and self-healing) properties of our solutions, both analytically and experimentally. The simulation results show that our solutions provide better performance in terms of coverage than pre-existing self-stabilizing algorithms

Fast and compact self-stabilizing verification, computation, and fault detection of an MST

This paper demonstrates the usefulness of distributed local verification of proofs, as a tool for the design of self-stabilizing algorithms.In particular, it introduces a somewhat generalized notion of distributed local proofs, and utilizes it for improving the time complexity significantly, while maintaining space optimality. As a result, we show that optimizing the memory size carries at most a small cost in terms of time, in the context of Minimum Spanning Tree (MST). That is, we present algorithms that are both time and space efficient for both constructing an MST and for verifying it.This involves several parts that may be considered contributions in themselves.First, we generalize the notion of local proofs, trading off the time complexity for memory efficiency. This adds a dimension to the study of distributed local proofs, which has been gaining attention recently. Specifically, we design a (self-stabilizing) proof labeling scheme which is memory optimal (i.e., $O(\log n)$ bits per node), and whose time complexity is $O(\log ^2 n)$ in synchronous networks, or $O(\Delta \log ^3 n)$ time in asynchronous ones, where $\Delta$ is the maximum degree of nodes. This answers an open problem posed by Awerbuch and Varghese (FOCS 1991). We also show that $\Omega(\log n)$ time is necessary, even in synchronous networks. Another property is that if $f$ faults occurred, then, within the requireddetection time above, they are detected by some node in the $O(f\log n)$ locality of each of the faults.Second, we show how to enhance a known transformer that makes input/output algorithms self-stabilizing. It now takes as input an efficient construction algorithm and an efficient self-stabilizing proof labeling scheme, and produces an efficient self-stabilizing algorithm. When used for MST, the transformer produces a memory optimal self-stabilizing algorithm, whose time complexity, namely, $O(n)$, is significantly better even than that of previous algorithms. (The time complexity of previous MST algorithms that used $\Omega(\log^2 n)$ memory bits per node was $O(n^2)$, and the time for optimal space algorithms was $O(n|E|)$.) Inherited from our proof labelling scheme, our self-stabilising MST construction algorithm also has the following two properties: (1) if faults occur after the construction ended, then they are detected by some nodes within $O(\log ^2 n)$ time in synchronous networks, or within $O(\Delta \log ^3 n)$ time in asynchronous ones, and (2) if $f$ faults occurred, then, within the required detection time above, they are detected within the $O(f\log n)$ locality of each of the faults. We also show how to improve the above two properties, at the expense of some increase in the memory