Deterministic Leader Election for Stationary Programmable Matter with Common Direction

TL;DR

This paper addresses deterministic leader election in stationary programmable matter on a triangular grid when all particles share a common direction but lack chirality and cannot move. It proves explicit termination is impossible and presents an implicitly terminating, movement-free LE that deterministically elects a single leader, employing boundary-based communication, boundary differentiation, and a tree-merge framework. The solution comprises Grey Leader Election on outer boundaries, Tree Construction to encode component structure, and Component Competition to merge trees until a single leader remains, with a total complexity of $O(n^3)$ activation rounds. The work highlights how restricting to a common direction yields different guarantees than chirality-based models and outlines open problems such as achieving explicit termination under restricted DBE settings and extending the approach to other PM tasks like shape formation.

Abstract

Leader Election is an important primitive for programmable matter, since it is often an intermediate step for the solution of more complex problems. Although the leader election problem itself is well studied even in the specific context of programmable matter systems, research on fault tolerant approaches is more limited. We consider the problem in the previously studied Amoebot model on a triangular grid, when the configuration is connected but contains nodes the particles cannot move to (e.g., obstacles). We assume that particles agree on a common direction (i.e., the horizontal axis) but do not have chirality (i.e., they do not agree on the other two directions of the triangular grid). We begin by showing that an election algorithm with explicit termination is not possible in this case, but we provide an implicitly terminating algorithm that elects a unique leader without requiring any movement. These results are in contrast to those in the more common model with chirality but no agreement on directions, where explicit termination is always possible but the number of elected leaders depends on the symmetry of the initial configuration. Solving the problem under the assumption of one common direction allows for a unique leader to be elected in a stationary and deterministic way, which until now was only possible for simply connected configurations under a sequential scheduler.

Figures (6)

• Figure 1: Example of grey, dark blue and light blue edges in two grey components connected by a DBE. The two grey components without any of the DBEs are traced in the second subfigure.
• Figure 2: Systems where all equivalent particles have the same local information cannot locally differentiate between the configurations.
• Figure 3: An example of cyclic--DFS demonstrating the traversal of a tree represented as a ring. The numbers denote the order in which each node is visited.
• Figure 4: Visualisation of the setting described in the proof of Lemma \ref{['lem:final-config-DBEs-internal']}
• Figure 5: A particle system consisting of $\frac{n}{2} + 2$ grey components and $n$ particles.

Theorems & Definitions (23)

• theorem 1
• proof
• theorem 2
• proof
• lemma 1
• proof
• lemma 2
• proof
• lemma 3
• proof