Conference Paper

# Self-Stabilization by Local Checking and Global Reset (Extended Abstract).

DOI: 10.1007/BFb0020443 Conference: Distributed Algorithms, 8th International Workshop, WDAG '94, Terschelling, The Netherlands, September 29 - October 1, 1994, Proceedings

Source: DBLP

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**ABSTRACT:**A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Yet despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. We consider the standard LOCAL model of computation and define LD(t) (for local decision) as the class of decision problems that can be solved in t communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD(t,p,q), containing all languages for which there exists a randomized algorithm that runs in t rounds, accepts correct instances with probability at least p, and rejects incorrect ones with probability at least q. We show that p2 + q = 1 is a threshold for the containment of LD(t) in BPLD(t,p,q). More precisely, we show that there exists a language that does not belong to LD(t) for any t=o(n) but does belong to BPLD(0,p,q) for any p,q ∈ (0,1) such that p2 + q ≤ 1. On the other hand, we show that, restricted to hereditary languages, BPLD(t,p,q)=LD(O(t)), for any function t, and any p, q ∈ (0,1) such that p2 + q > 1. In addition, we investigate the impact of nondeterminism on local decision, and establish several structural results inspired by classical computational complexity theory. Specifically, we show that nondeterminism does help, but that this help is limited, as there exist languages that cannot be decided locally nondeterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with nondeterminism that enables to decide all languages in constant time. Finally, we introduce the notion of local reduction, and establish a couple of completeness results.Journal of the ACM (JACM). 10/2013; 60(5). - [Show abstract] [Hide abstract]

**ABSTRACT:**Developing self-stabilizing solutions is considered to be more challenging and complicated than developing classical solutions, where a proper initialization of the variables can be assumed. Hence, to ease the task of the developers, some automatic techniques have been proposed to design self-stabilizing algorithms. In this paper, we propose an automatic transformer for algorithms in an extended population protocol model. Population protocols is a model that was introduced recently for networks with a large number of resource-limited mobile agents. We use a variant of this model. First, we assume agents having characteristics (e.g., moving speed, communication radius) affecting their intercommunication “speed”, which is reflected by their cover times. Second, we assume the existence of a special agent with an unbounded memory, the base station. The automatic transformer takes as an input an algorithm solving a static problem (and meeting some additional rather natural requirements) and outputs a self-stabilizing algorithm for the same problem. The transformer is built using a re-execution approach (the technique consisting of executing an algorithm repeatedly in order to obtain its self-stabilizing version). We show that in the model we use, a transformer based on such an approach is impossible without the assumption of an unbounded memory agent.Theoretical Computer Science 01/2011; · 0.49 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**A fault-tolerant and stabilizing simulation of an atomic register is presented. The simulation works in asynchronous message-passing systems, and allows a minority of processes to crash. The simulation stabilizes in a pragmatic manner, by reaching a long execution in which it runs correctly. A key element in the simulation is a new combinatorial construction of a bounded labeling scheme accommodating arbitrary labels, including those not generated by the scheme itself.Stabilization, Safety, and Security of Distributed Systems - 13th International Symposium, SSS 2011, Grenoble, France, October 10-12, 2011. Proceedings; 01/2011

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