## 抄録

Shape formation (or pattern formation) is a basic distributed problem for systems of computational mobile entities. Intensively studied for systems of autonomous mobile robots, it has recently been investigated in the realm of programmable matter, where entities are assumed to be small and with severely limited capabilities. Namely, it has been studied in the geometric Amoebot model, where the anonymous entities, called particles, operate on a hexagonal tessellation of the plane and have limited computational power (they have constant memory), strictly local interaction and communication capabilities (only with particles in neighboring nodes of the grid), and limited motorial capabilities (from a grid node to an empty neighboring node); their activation is controlled by an adversarial scheduler. Recent investigations have shown how, starting from a well-structured configuration in which the particles form a (not necessarily complete) triangle, the particles can form a large class of shapes. This result has been established under several assumptions: agreement on the clockwise direction (i.e., chirality), a sequential activation schedule, and randomization (i.e., particles can flip coins to elect a leader). In this paper we provide a characterization of which shapes can be formed deterministically starting from any simply connected initial configuration of n particles. The characterization is constructive: we provide a universal shape formation algorithm that, for each feasible pair of shapes (S_{0}, S_{F}), allows the particles to form the final shape S_{F} (given in input) starting from the initial shape S_{0}, unknown to the particles. The final configuration will be an appropriate scaled-up copy of S_{F} depending on n. If randomization is allowed, then any input shape can be formed from any initial (simply connected) shape by our algorithm, provided that there are enough particles. Our algorithm works without chirality, proving that chirality is computationally irrelevant for shape formation. Furthermore, it works under a strong adversarial scheduler, not necessarily sequential. We also consider the complexity of shape formation both in terms of the number of rounds and the total number of moves performed by the particles executing a universal shape formation algorithm. We prove that our solution has a complexity of O(n^{2}) rounds and moves: this number of moves is also asymptotically worst-case optimal.

本文言語 | 英語 |
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ホスト出版物のタイトル | 21st International Conference on Principles of Distributed Systems, OPODIS 2017 |

編集者 | James Aspnes, Joao Leitao, Alysson Bessani, Pascal Felber |

出版社 | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN（電子版） | 9783959770613 |

DOI | |

出版ステータス | 出版済み - 3月 1 2018 |

イベント | 21st International Conference on Principles of Distributed Systems, OPODIS 2017 - Lisboa, ポルトガル 継続期間: 12月 18 2017 → 12月 20 2017 |

### 出版物シリーズ

名前 | Leibniz International Proceedings in Informatics, LIPIcs |
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巻 | 95 |

ISSN（印刷版） | 1868-8969 |

### その他

その他 | 21st International Conference on Principles of Distributed Systems, OPODIS 2017 |
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国/地域 | ポルトガル |

City | Lisboa |

Period | 12/18/17 → 12/20/17 |

## !!!All Science Journal Classification (ASJC) codes

- ソフトウェア