Efficient Deterministic Leader Election for Programmable Matter

TL;DR

This paper introduces a deterministic leader election algorithm for programmable matter that allows temporary disconnection and reconnects efficiently, achieving linear runtime comparable to randomized algorithms and faster than previous deterministic methods.

Contribution

The paper presents a novel deterministic leader election algorithm enabling temporary disconnection, achieving linear runtime, and outperforming previous deterministic algorithms in general shapes.

Findings

Runtime is linear in the diameter of the system.

The algorithm matches the speed of randomized algorithms.

It improves upon previous deterministic algorithms with quadratic or worse runtime.

Abstract

It was suggested that a programmable matter system (composed of multiple computationally weak mobile particles) should remain connected at all times since otherwise, reconnection is difficult and may be impossible. At the same time, it was not clear that allowing the system to disconnect carried a significant advantage in terms of time complexity. We demonstrate for a fundamental task, that of leader election, an algorithm where the system disconnects and then reconnects automatically in a non-trivial way (particles can move far away from their former neighbors and later reconnect to others). Moreover, the runtime of the temporarily disconnecting deterministic leader election algorithm is linear in the diameter. Hence, the disconnecting -- reconnecting algorithm is as fast as previous randomized algorithms. When comparing to previous deterministic algorithms, we note that some of the previous work assumed weaker schedulers. Still, the runtime of all the previous deterministic algorithms that did not assume special shapes of the particle system (shapes with no holes) was at least quadratic in $n$, where $n$ is the number of particles in the system. (Moreover, the new algorithm is even faster in some parameters than the deterministic algorithms that did assume special initial shapes.) Since leader election is an important module in algorithms for various other tasks, the presented algorithm can be useful for speeding up other algorithms under the assumption of a strong scheduler. This leaves open the question: "can a deterministic algorithm be as fast as the randomized ones also under weaker schedulers?"