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

Jérémie Chalopin, Shantanu Das, Maria Kokkou

TL;DR

The paper tackles deterministic leader election in programmable matter with constant-memory particles by introducing a silent self-stabilising algorithm tailored to simply connected 2D triangular-grid configurations. It combines a constant-memory proof-labelling certificate (edge orientation with limited outgoing edges and no directed triangles) with a simple self-stabilising edge-orientation algorithm, proving convergence to a configuration with a unique sink (the leader) under Gouda fair scheduling. A minimal counterexample proof structure, leveraging boundary geometry and several lemmas on boundary particles, establishes correctness of both the certificate and the algorithm. The work highlights the feasibility of constant-memory, self-stabilising leader election in geometry-rich programmable matter and motivates future directions for scheduler assumptions and broader graph classes.

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 $Ω(\log{n})$ bits of memory per node, even for ring topologies [Dolev et al. 1999].

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

TL;DR

The paper tackles deterministic leader election in programmable matter with constant-memory particles by introducing a silent self-stabilising algorithm tailored to simply connected 2D triangular-grid configurations. It combines a constant-memory proof-labelling certificate (edge orientation with limited outgoing edges and no directed triangles) with a simple self-stabilising edge-orientation algorithm, proving convergence to a configuration with a unique sink (the leader) under Gouda fair scheduling. A minimal counterexample proof structure, leveraging boundary geometry and several lemmas on boundary particles, establishes correctness of both the certificate and the algorithm. The work highlights the feasibility of constant-memory, self-stabilising leader election in geometry-rich programmable matter and motivates future directions for scheduler assumptions and broader graph classes.

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 bits of memory per node, even for ring topologies [Dolev et al. 1999].
Paper Structure (8 sections, 10 theorems, 4 equations, 7 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 10 theorems, 4 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our results
  3. Related Work
  4. Model
  5. A Proof Labelling Scheme for Leader Election
  6. A Self-Stabilising Algorithm for Leader Election
  7. Proof of Theorem \ref{['th:certificate']} and Theorem \ref{['th:algo-correct']}
  8. Further Remarks

Key Result

Lemma 1

In any simply connected particle system $\mathcal{P}\xspace$ with at least two particles, the boundary of $\mathcal{P}\xspace$ contains one of the following:

Figures (7)

  • Figure 1: An 18-particle ring where for each particle, the occupied neighbours are reached through port numbers 2 and 4. Two nodes have the same colour if they agree on the grid orientation.
  • Figure 2: The 1-view of $p$ and the two possible orientations of $q$
  • Figure 3: A particle configuration satisfying the first and last condition of Claim \ref{['claim:chirality']}, but not the second.
  • Figure 4: A configuration containing broken rules. Each square particle can detect at least one rule being violated. Green (dashed) edges break \ref{['rule:all-edges-directed']}, grey edges break \ref{['rule:incoming-edge']}, black (dash dotted) edges break \ref{['rule:consecutive']} and blue edges break \ref{['rule:no-cyclic-triangles']}.
  • Figure 5: An example of a configuration containing a directed cycle (marked by blue edges) of more than three particles that does not violate any rule.
  • ...and 2 more figures

Theorems & Definitions (36)

  • Remark 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Claim 1
  • proof
  • Theorem 1
  • Theorem 2
  • ...and 26 more