Making Self-Stabilizing Algorithms for Any Locally Greedy Problem

Self-stabilizing algorithms are a way to deal with network dynamicity, as it will update itself after a network change (addition or removal of nodes or edges), as long as changes are not frequent. We propose an automatic transformation of synchronous distributed algorithms that solve locally greedy and mendable problems into self-stabilizing algorithms in anonymous networks. Mendable problems are a generalization of greedy problems where any partial solution may be transformed -instead of completed- into a global solution: every time we extend the partial solution, we are allowed to change the previous partial solution up to a given distance. Locally here means that to extend a solution for a node, we need to look at a constant distance from it. In order to do this, we propose the first explicit self-stabilizing algorithm computing a (k,k-1)-ruling set (i.e. a "maximal independent set at distance k"). By combining this technique multiple times, we compute a distance-K coloring of the graph. With this coloring we can finally simulate Local model algorithms running in a constant number of rounds, using the colors as unique identifiers. Our algorithms work under the Gouda daemon, similar to the probabilistic daemon: if an event should eventually happen, it will occur

Making Self-Stabilizing any Locally Greedy Problem

We propose a way to transform synchronous distributed algorithms solving locally greedy and mendable problems into self-stabilizing algorithms in anonymous networks. Mendable problems are a generalization of greedy problems where any partial solution may be transformed -- instead of completed -- into a global solution: every time we extend the partial solution we are allowed to change the previous partial solution up to a given distance. Locally here means that to extend a solution for a node, we need to look at a constant distance from it. In order to do this, we propose the first explicit self-stabilizing algorithm computing a $(k,k-1)$-ruling set (i.e. a "maximal independent set at distance $k$"). By combining multiple time this technique, we compute a distance-$K$ coloring of the graph. With this coloring we can finally simulate \local~model algorithms running in a constant number of rounds, using the colors as unique identifiers. Our algorithms work under the Gouda daemon, which is similar to the probabilistic daemon: if an event should eventually happen, it will occur under this daemon

Self-stabilizing algorithms for Connected Vertex Cover and Clique decomposition problems

In many wireless networks, there is no fixed physical backbone nor centralized network management. The nodes of such a network have to self-organize in order to maintain a virtual backbone used to route messages. Moreover, any node of the network can be a priori at the origin of a malicious attack. Thus, in one hand the backbone must be fault-tolerant and in other hand it can be useful to monitor all network communications to identify an attack as soon as possible. We are interested in the minimum \emph{Connected Vertex Cover} problem, a generalization of the classical minimum Vertex Cover problem, which allows to obtain a connected backbone. Recently, Delbot et al.~\cite{DelbotLP13} proposed a new centralized algorithm with a constant approximation ratio of $2$ for this problem. In this paper, we propose a distributed and self-stabilizing version of their algorithm with the same approximation guarantee. To the best knowledge of the authors, it is the first distributed and fault-tolerant algorithm for this problem. The approach followed to solve the considered problem is based on the construction of a connected minimal clique partition. Therefore, we also design the first distributed self-stabilizing algorithm for this problem, which is of independent interest

Efficient self-stabilizing construction of disjoint MDSs in distance-2 model

We study the deterministic silent self-stabilizing construction of two disjoint minimal dominating sets (MDSs) in anonymous networks. We focus on algorithms where nodes share only their status (i.e. the name of their MDS to which they belong, if they belong to a MDS). We prove that such an algorithm cannot be designed in distance-1 model under a central daemon; therefore, we study this problem in the distance-2 model under a central daemon. We present an algorithm building two disjoint minimal dominating sets such that one of them is also a maximal independent set (MIS). Any execution of this algorithm converges in 5n moves. Our approach to compute this value is novel: the number of moves is not computed per node. We propose a second algorithm faster than the first one at the expense of the independence property of one of the constructed sets. A node executes at most 2 moves. If the network is not anonymous, the presented algorithms can be translated into a silent self-stabilizing algorithms converging in O($n$•$m$) moves in the distance-1 model under the distributed daemon where m is the number of edges and n the number of nodes. This improves the complexity of O($n$.$m$) moves of proposed algorithms with the same assumptions

Automated Synthesis of Distributed Self-Stabilizing Protocols

In this paper, we introduce an SMT-based method that automatically synthesizes a distributed self-stabilizing protocol from a given high-level specification and network topology. Unlike existing approaches, where synthesis algorithms require the explicit description of the set of legitimate states, our technique only needs the temporal behavior of the protocol. We extend our approach to synthesize ideal-stabilizing protocols, where every state is legitimate. We also extend our technique to synthesize monotonic-stabilizing protocols, where during recovery, each process can execute an most once one action. Our proposed methods are fully implemented and we report successful synthesis of well-known protocols such as Dijkstra's token ring, a self-stabilizing version of Raymond's mutual exclusion algorithm, ideal-stabilizing leader election and local mutual exclusion, as well as monotonic-stabilizing maximal independent set and distributed Grundy coloring

Communication Efficiency in Self-stabilizing Silent Protocols

Self-stabilization is a general paradigm to provide forward recovery capabilities to distributed systems and networks. Intuitively, a protocol is self-stabilizing if it is able to recover without external intervention from any catastrophic transient failure. In this paper, our focus is to lower the communication complexity of self-stabilizing protocols \emph{below} the need of checking every neighbor forever. In more details, the contribution of the paper is threefold: (i) We provide new complexity measures for communication efficiency of self-stabilizing protocols, especially in the stabilized phase or when there are no faults, (ii) On the negative side, we show that for non-trivial problems such as coloring, maximal matching, and maximal independent set, it is impossible to get (deterministic or probabilistic) self-stabilizing solutions where every participant communicates with less than every neighbor in the stabilized phase, and (iii) On the positive side, we present protocols for coloring, maximal matching, and maximal independent set such that a fraction of the participants communicates with exactly one neighbor in the stabilized phase

Optimal Dynamic Distributed MIS

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees. The result trivially applies for the dynamic distributed model, in which edges or nodes may be inserted or deleted. In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either \emph{synchronous} or \emph{asynchronous}, with only \emph{a single adjustment} and in a single round, in expectation. These strong guarantees hold for the \emph{complete fully dynamic} setting: Insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former. Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy \emph{sequential} MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a $3$-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing. Finally, our algorithm enjoys a useful \emph{history-independence} property, meaning the output is independent of the history of topology changes that constructed that graph. This means the output cannot be chosen, or even biased, by the adversary in case its goal is to prevent us from optimizing some objective function.Comment: 19 pages including appendix and reference