🤖 AI Summary
This paper addresses the unique leader election problem for programmable matter systems modeled as two-dimensional triangular grids, under the constraint of constant memory (O(1) bits per particle) and without any initial state assumptions. We propose the first silent, deterministic, self-stabilizing, and resident distributed algorithm for this setting—breaking the classical Ω(log n) memory lower bound for leader election in general distributed systems. Our approach explicitly exploits the geometric symmetry and local neighborhood structure of the triangular grid to circumvent inherent limitations of purely graph-theoretic models. The algorithm operates under Gouda’s fair scheduler model and employs a geometry-aware state-machine synchronization mechanism, guaranteeing convergence to a unique-leader configuration from any arbitrary initial state. We formally prove its correctness, self-stabilization, and silence. This work establishes a scalable, low-overhead primitive for large-scale, energy-constrained programmable matter systems.
📝 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].