Near-Optimal Self-Stabilising Counting and Firing Squads

TL;DR

This paper introduces a framework for solving self-stabilising counting and firing squad problems in distributed systems, achieving near-optimal resilience, stabilisation, and response times by reducing these tasks to binary consensus.

Contribution

The paper presents a novel framework that reduces self-stabilising counting and firing squad problems to binary consensus, enabling efficient deterministic and randomized solutions with optimal resilience.

Findings

Deterministic algorithm for Byzantine firing squads with resilience f<n/3

Asymptotically optimal stabilisation and response time O(f)

Message size of O(log f)

Abstract

Consider a fully-connected synchronous distributed system consisting of $n$ nodes, where up to $f$ nodes may be faulty and every node starts in an arbitrary initial state. In the synchronous $C$-counting problem, all nodes need to eventually agree on a counter that is increased by one modulo $C$ in each round for given $C>1$. In the self-stabilising firing squad problem, the task is to eventually guarantee that all non-faulty nodes have simultaneous responses to external inputs: if a subset of the correct nodes receive an external "go" signal as input, then all correct nodes should agree on a round (in the not-too-distant future) in which to jointly output a "fire" signal. Moreover, no node should generate a "fire" signal without some correct node having previously received a "go" signal as input. We present a framework reducing both tasks to binary consensus at very small cost. For example, we obtain a deterministic algorithm for self-stabilising Byzantine firing squads with optimal resilience $f<n/3$, asymptotically optimal stabilisation and response time $O(f)$, and message size $O(\log f)$. As our framework does not restrict the type of consensus routines used, we also obtain efficient randomised solutions, and it is straightforward to adapt our framework for other types of permanent faults.