Randomized self-stabilizing and space optimal leader election under arbitrary scheduler on rings

Abstract

We present a randomized self-stabilizing leader election protocol and a randomized self-stabilizing token circulation protocol under an arbitrary scheduler on anonymous and unidirectional rings of any size. These protocols are space optimal. We also give a formal and complete proof of these protocols. To this end, we develop a complete model for probabilistic self-stabilizing distributed systems which clearly separates the non deterministic behavior of the scheduler from the randomized behavior of the protocol. This framework includes all the necessary tools for proving the selfstabilization of a randomized distributed system: definition of a probabilistic space and definition of the self-stabilization of a randomized protocol. We also propose a new technique of scheduler management through a self-stabilizing protocol composition (cross-over composition). Roughly speaking, we force all computations to have a fairness property under any scheduler, even under an unfair one.

Robust sensor self-initialization: Whispering to avoid intruders

“…In such a situation the collision probability (i.e. the probability of two nodes choosing the same ID) is given by the birthday paradox p(X(n t )) = 1 − 16 ), in which X is a discrete uniform distribution of IDs and n t is the total number of nodes deployed. As it can be seen in Figure 1, the collision probability is over 10% with only 120 nodes, and reaches 50% with ∼ 300 nodes.…”

“…We have a much more relaxed goal. We want to find a 2-distance graph coloring with a minimum number of messages, bounded by a maximum of 2 16 colors.…”

“…[16] redefined self-stabilization for probabilistic protocols in such a way that regardless of the initial state of the system and regardless of the scheduler strategy, the system will reach a legitimate final state with probability 1. Using the framework for proving self-stabilization of probabilistic protocols, defined in [16], it can be shown that the previously described protocol, satisfies the above mentioned definition of self-stabilization, if extended IDs are used to link queries and replies.…”

“…Informally, the framework, defined in [16], states that a probabilistic protocol is self-stabilizing for a given specification if there is a sequence of predicates over system states L i (S) . .…”

Leader Election in Anonymous Rings: Franklin Goes Probabilistic

“…Angluin, Aspnes, Fischer and Jiang [2] construct a self-stabilizing leader election algorithm for anonymous, unidirectional, asynchronous rings of odd size in the framework of their model of population protocols. Beauquier, Gradinariu and Johnen [5] present a randomized self-stabilizing leader election algorithm under an arbitrary scheduler (no fairness assumption is required) on anonymous, unidirectional rings of known size, in the shared variables model. Both algorithms are based on token circulation.…”