Dijkstra's Self-Stabilizing Algorithm in Unsupportive Environments

TL;DR

This paper extends Dijkstra's self-stabilizing algorithm to environments with weaker communication atomicity, including regular and safe registers, and introduces probabilistic register models requiring weak stabilization.

Contribution

It introduces versions of Dijkstra's algorithm that operate under weaker atomicity assumptions and proposes a probabilistic register implementation with weak stabilization.

Findings

Solutions for regular registers are provided.

Negative and positive results for safe registers are discussed.

A probabilistic register implementation with weak stabilization is presented.

Abstract

The first self-stabilizing algorithm [Dij73] assumed the existence of a central daemon, that activates one processor at time to change state as a function of its own state and the state of a neighbor. Subsequent research has reconsidered this algorithm without the assumption of a central daemon, and under different forms of communication, such as the model of link registers. In all of these investigations, one common feature is the atomicity of communication, whether by shared variables or read/write registers. This paper weakens the atomicity assumptions for the communication model, proposing versions of [Dij73] that tolerate various weaker forms of atomicity. First, a solution for the case of regular registers is presented. Then the case of safe registers is considered, with both negative and positive results presented. The paper also presents an implementation of [Dij73] based on registers that have probabilistically correct behavior, which requires a notion of weak stabilization.