Silent Self-Stabilising Leader Election in Programmable Matter Systems with Holes

TL;DR

It is shown that movement in conjunction with particles sharing a sense of orientation and operating in a grid can overcome classical impossibility results for constant-memory systems established by Dolev, Gouda and Schneider (1999).

Abstract

Leader election is a fundamental problem in distributed computing, particularly within programmable matter systems, where coordination among simple computational entities is crucial for solving complex tasks. In these systems, particles (i.e., constant-memory computational entities) operate in a regular triangular grid as described in the geometric Amoebot model. While leader election has been extensively studied in non self-stabilising settings, self-stabilising solutions remain more limited. In this work, we study the problem of self-stabilising leader election in connected (but not necessarily simply connected) configurations. We present the first self-stabilising algorithm for connected programmable matter systems that guarantees the election of a unique leader under an unfair scheduler, for oblivious particles (i.e., particles with no persistent memory) that share a common sense of direction. Our approach leverages particle movement, a capability not previously exploited in the self-stabilising context. We show that movement in conjunction with particles sharing a sense of orientation and operating in a grid can overcome classical impossibility results for constant-memory systems established by Dolev, Gouda and Schneider (1999).

Figures (8)

• Figure 1: A particle configuration where all particles share identical local information. Blue and black particles have opposite chiralities (i.e., senses of rotational orientation). Numbers denote (consecutive) edge labels from 0 to 5.
• Figure 2: Directions of particles
• Figure 3: An expanded particle (symbolised by two circles connected with a line) that cannot contract without disconnecting its neighbours (depicted as black circles). The tail is denoted by $t$ and the head by $h$. Nodes marked by x are assumed to be empty.
• Figure 4: (a) An example configuration containing particles that satisfy all conditions. The blue expanded particle satisfies \ref{['cond:expanded-contract']}, the black square expanded particle satisfies \ref{['cond:expand-lower-head']}, the white circle expanded particle satisfies \ref{['cond:expand-lower-tail']}, the white square expanded particle satisfies \ref{['cond:expand-other-diagonal']}, the blue circle contracted particle satisfies \ref{['cond:contracted-expand']} and the grey contracted particle satisfies \ref{['cond:contracted-lowermost-right']}. (b) A configuration where the only satisfied condition is \ref{['cond:expand-other-diagonal']} (for the pink expanded particle).
• Figure 5: Black circles represent particles. Two black circles connected with a line represent an expanded particle. White circles can be empty or occupied by particles. Nodes marked by x are assumed to be empty.

Theorems & Definitions (9)

• proposition 1
• lemma 1
• proof
• proof : Proof of Proposition \ref{['prop:final']}
• lemma 2
• proof
• lemma 3
• proof
• theorem 1