A Markov Chain Algorithm for Compression in Self-Organizing Particle Systems

Abstract

We consider programmable matter as a collection of simple computational elements (or particles) with limited (constantsize) memory that self-organize to solve system-wide problems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the particle system gathers as tightly together as possible, as in a sphere or its equivalent in the presence of some underlying geometry. More specifically, we seek fully distributed, local, and asynchronous algorithms that lead the system to converge to a configuration with small perimeter. We present a Markov chain based algorithm that solves the compression problem under the geometric amoebot model, for particle systems that begin in a connected configuration with no holes. The algorithm takes as input a bias parameter λ, where λ > 1 corresponds to particles favoring inducing more lattice triangles within the particle system. We show that for all λ > 5, there is a constant α > 1 such that at stationarity with all but exponentially small probability the particles are α-compressed, meaning the perimeter of the system configuration is at most α • pmin, where pmin is the minimum possible perimeter of the particle system. We additionally prove that the same algorithm can be used for expansion for small values of λ; in particular, for all 0 < λ < √ 2, there is a constant β < 1 such that at stationarity, with all but an ex-* A full version of this paper, including omitted proofs, is

A Stochastic Approach to Shortcut Bridging in Programmable Matter

“…In [5], we introduced a stochastic, distributed algorithm for compression in the amoebot model; here we extend that work to show that stochastic approach is in fact more generally applicable. The motivation underlying this Markov chain approach to programmable matter comes from statistical physics, where ensembles of particles reminiscent of the amoebot model are used to study physical systems and demonstrate that local micro-behavior can induce global macro-scale changes to the system [2,23,3].…”

“…We recall the main properties of the amoebot model [5,12], an abstract model for programmable matter that provides a framework for rigorous algorithmic research on nano-scale systems. We represent programmable matter as a collection of individual computational units known as particles.…”

“…Compared to other algorithms for programmable matter and self-organizing particle systems, stochastic methods such as the compression algorithm of [5] and our shortcut bridging algorithm are nearly oblivious, more robust to failures, and require little to no communication between particles. Because each of these algorithms is derived from a stochastic process, powerful tools developed for Markov chain analysis can be employed to rigorously understand their behavior.…”

“…Our work presents a plausible explanation of how convergence to globally optimal configurations can be achieved via local interactions by simple organisms (e.g., ants) with some limited computational power and access to random bits. The proposed algorithm demonstrates the robustness of the stochastic approach to algorithms for programmable matter, as it is a surprisingly simple extension of a stochastic algorithm for compression [5]. …”

“…Our proposed algorithm for shortcut bridging is an extension of the stochastic, distributed algorithm for the compression problem introduced in [5], which shows that many fundamental elements of our stochastic approach can be generalized to applications beyond the specific context of compression. In particular, our stochastic approach may be of future interest in the molecular programming domain, where simpler variations of bridging have been studied.…”