### Abstract

We consider programmable matter as a collection of simple computational elements (or particles) with limited (constant- size) memory that self-organize to solve system-wide prob- lems of movement, configuration, and coordination. Here, we focus on the compression problem, in which the parti- cle 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 con- verge 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 sys- tems 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 lat- tice triangles within the particle system. We show that for all λ > 5, there is a constant α > 1 such that at stationar- ity with all but exponentially small probability the particles are α-compressed, meaning the perimeter of the system con- figuration is at most α · p_{min}, where p_{min} 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 fi < 1 such that at stationarity, with all but an ex-ponentially small probability, the perimeter will be at least β · p_{max}, where p_{max} is the maximum possible perimeter.

Original language | English (US) |
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Title of host publication | PODC 2016 - Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing |

Publisher | Association for Computing Machinery |

Pages | 279-288 |

Number of pages | 10 |

Volume | 25-28-July-2016 |

ISBN (Electronic) | 9781450339643 |

DOIs | |

State | Published - Jul 25 2016 |

Event | 35th ACM Symposium on Principles of Distributed Computing, PODC 2016 - Chicago, United States Duration: Jul 25 2016 → Jul 28 2016 |

### Other

Other | 35th ACM Symposium on Principles of Distributed Computing, PODC 2016 |
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Country | United States |

City | Chicago |

Period | 7/25/16 → 7/28/16 |

### Keywords

- Compression
- Markov Chains
- Self-organizing Particles

### ASJC Scopus subject areas

- Software
- Hardware and Architecture
- Computer Networks and Communications

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## Cite this

*PODC 2016 - Proceedings of the 2016 ACM Symposium on Principles of Distributed Computing*(Vol. 25-28-July-2016, pp. 279-288). Association for Computing Machinery. https://doi.org/10.1145/2933057.2933107