Deterministic Self-Stabilising Leader Election for Programmable Matter with Constant Memory

TL;DR

This paper introduces a simple, deterministic, silent self-stabilising leader election algorithm for programmable matter particles with constant memory, leveraging geometric properties of 2D triangular grids to achieve stabilization.

Contribution

It presents the first self-stabilising leader election algorithm for programmable matter with constant memory, exploiting geometric properties of 2D triangular grids.

Findings

Algorithm always stabilizes to a unique leader

Requires constant memory per particle

Works under fairness assumptions (Gouda fairness)

Abstract

The problem of electing a unique leader is central to all distributed systems, including programmable matter systems where particles have constant size memory. In this paper, we present a silent self-stabilising, deterministic, stationary, election algorithm for particles having constant memory, assuming that the system is simply connected. Our algorithm is elegant and simple, and requires constant memory per particle. We prove that our algorithm always stabilises to a configuration with a unique leader, under a daemon satisfying some fairness guarantees (Gouda fairness [Gouda 2001]). We use the special geometric properties of programmable matter in 2D triangular grids to obtain the first self-stabilising algorithm for such systems. This result is surprising since it is known that silent self-stabilising algorithms for election in general distributed networks require $\Omega(\log{n})$ bits of memory per node, even for ring topologies [Dolev et al. 1999].