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

Cannon, Sarah; Daymude, Joshua J.; Gokmen, Cem; Randall, Dana; Richa, Andréa W.

doi:10.48550/arxiv.1805.04599

Cited by 1 publication

(19 citation statements)

References 24 publications

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“…Here we generalize the bridging techniques to account for more complex contours that form an interconnected network to show that the contour lengths of the bridging system can be made arbitrarily close to their minimum possible length and, as a result, alignment occurs with high probability. We note that our algorithms for alignment in both settings work for all q ≥ 2; separation (where the sizes of the color classes are fixed) has only been shown for q = 2, although the methods should also generalize to more colors [6].…”

Section: Figurementioning

confidence: 95%

“…Configurations with two dominant orientations (black vs. gray circles); large interfaces as in (a) are unlikely for large γ, whereas small interfaces as in (b) are likely for any finite γ. partition function" in terms of the volume and surface contributions, as in [5,6,14], to prove that our algorithms achieve compression (or aggregation), with high probability. Moreover, using isoperimetric inequalities, we prove the absence of bottlenecks in sufficiently highly compressed configurations, which is necessary to get the system to globally align.…”

Section: Figurementioning

confidence: 99%

“…Moreover, using isoperimetric inequalities, we prove the absence of bottlenecks in sufficiently highly compressed configurations, which is necessary to get the system to globally align. Finally, we use the bridging techniques first proposed in [27] and later adapted in [5,6], to expand the information theoretic arguments in [5,6] to prove that for sufficiently compressed configurations, our algorithms achieve alignment with high probability. Conversely, we show that our algorithms can achieve expansion and/or non-alignment (with all directions nearly equitably balanced), with the same algorithm by adjusting only two global parameters.…”

Section: Figurementioning

confidence: 99%

“…Our model of programmable matter is based on the amoebot model, introduced in [11] and described in detail in [10], which has served as the basis for previous stochastic algorithms for SOPS [8,7,6,5]. In the amoebot model, particles occupy the nodes of a graph with each node occupied by at most one particle.…”

Section: Preliminariesmentioning

confidence: 99%

“…We consider a stochastic approach, used previously in [7,8] to achieve compression, where connected sets of homogeneous particles self-organize to gather together tightly, separation in heterogeneous particle systems, where all of the particles compress, but also gather most tightly with other particles of the same type [5,6], and aggregation of homogeneous particles that are not required to be connected, where most particles accumulate in a small, compact neighborhood [21]. In all of these, phase changes were used to characterize desirable behaviors at stationarity, with high probability.…”

Section: Introductionmentioning

confidence: 99%

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Local Stochastic Algorithms for Alignment in Self-Organizing Particle Systems

Kedia¹,

Oh²,

Randall³

2022

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We present local distributed, stochastic algorithms for alignment in self-organizing particle systems (SOPS) on two-dimensional lattices, where particles occupy unique sites on the lattice, and particles can make spatial moves to neighboring sites if they are unoccupied. Such models are abstractions of programmable matter, composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational capabilities. We consider oriented particle systems, where particles are assigned a vector pointing in one of q directions, and each particle can compute the angle between its direction and the direction of any neighboring particle, although without knowledge of global orientation with respect to a fixed underlying coordinate system. Particles move stochastically, with each particle able to either modify its direction or make a local spatial move along a lattice edge during a move. We consider two settings: (a) where particle configurations must remain simply connected at all times and (b) where spatial moves are unconstrained and configurations can disconnect.Our algorithms are inspired by the Potts model and its planar oriented variant known as the planar Potts model or clock model from statistical physics. We prove that for any q ≥ 2, by adjusting a single parameter, these self-organizing particle systems can be made to collectively align along a single dominant direction (analogous to a solid or ordered state) or remain non-aligned, in which case the fraction of particles oriented along any direction is nearly equal (analogous to a gaseous or disordered state). In the connected SOPS setting, we allow for two distinct parameters, one controlling the ferromagnetic attraction between neighboring particles (regardless of orientation) and the other controlling the preference of neighboring particles to align. We show that with appropriate settings of the input parameters, we can achieve compression and expansion, controlling how tightly gathered the particles are, as well as alignment or nonalignment, producing a single dominant orientation or not. While alignment is known for the Potts and clock models at sufficiently low temperatures, our proof in the SOPS setting are significantly more challenging because the particles make spatial moves, not all sites are occupied, and the total number of particles is fixed. ACM Subject ClassificationMathematics of computing → Stochastic processes; Theory of computation → Self-organization; Theory of computation → Random walks and Markov chains Keywords and phrases Self-organizing particle systems, alignment, Markov chains, active matter.

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Section: Figurementioning

confidence: 95%

Section: Figurementioning

confidence: 99%