### 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) that 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.

Original language | English (US) |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019 |

Editors | Dimitris Achlioptas, Laszlo A. Vegh |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771252 |

DOIs | |

State | Published - Sep 2019 |

Externally published | Yes |

Event | 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 - Cambridge, United States Duration: Sep 20 2019 → Sep 22 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 145 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 |
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Country | United States |

City | Cambridge |

Period | 9/20/19 → 9/22/19 |

### Fingerprint

### Keywords

- Cluster expansion
- Markov chains
- Programmable matter

### ASJC Scopus subject areas

- Software

### Cite this

*Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019*[54] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 145). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.54

**A local stochastic algorithm for separation in heterogeneous self-organizing particle systems.** / Cannon, Sarah; Daymude, Joshua J.; Gökmen, Cem; Randall, Dana; Richa, Andréa W.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019.*, 54, Leibniz International Proceedings in Informatics, LIPIcs, vol. 145, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019, Cambridge, United States, 9/20/19. https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.54

}

TY - GEN

T1 - A local stochastic algorithm for separation in heterogeneous self-organizing particle systems

AU - Cannon, Sarah

AU - Daymude, Joshua J.

AU - Gökmen, Cem

AU - Randall, Dana

AU - Richa, Andréa W.

PY - 2019/9

Y1 - 2019/9

N2 - 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) that 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.

AB - 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) that 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.

KW - Cluster expansion

KW - Markov chains

KW - Programmable matter

UR - http://www.scopus.com/inward/record.url?scp=85072865042&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072865042&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX-RANDOM.2019.54

DO - 10.4230/LIPIcs.APPROX-RANDOM.2019.54

M3 - Conference contribution

AN - SCOPUS:85072865042

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019

A2 - Achlioptas, Dimitris

A2 - Vegh, Laszlo A.

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -