A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems

TL;DR

This paper introduces and analyzes a distributed stochastic algorithm enabling heterogeneous self-organizing particle systems to either separate by color or integrate, influenced by a global parameter, with rigorous proofs of behavior.

Contribution

It extends a previous homogeneous algorithm to heterogeneous systems, providing a rigorous analysis and new techniques for proving separation and integration behaviors.

Findings

The algorithm successfully achieves separation or integration based on a global parameter.

The analysis employs advanced techniques from statistical physics and probabilistic methods.

The approach generalizes previous work on homogeneous systems to heterogeneous, bichromatic systems.

Abstract

We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC '16), which can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM '11), the high-temperature expansion of the Ising model, and careful probabilistic arguments.