Shape formation by programmable particles

Giuseppe A. Di Luna, Paola Flocchini, Nicola Santoro, Giovanni Viglietta, Yukiko Yamauchi

Research output: Chapter in Book/Report/Conference proceedingConference contribution

10 Citations (Scopus)

Abstract

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 (S0, SF), allows the particles to form the final shape SF (given in input) starting from the initial shape S0, unknown to the particles. The final configuration will be an appropriate scaled-up copy of SF 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(n2) rounds and moves: this number of moves is also asymptotically worst-case optimal.

Original languageEnglish
Title of host publication21st International Conference on Principles of Distributed Systems, OPODIS 2017
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Volume95
ISBN (Electronic)9783959770613
DOIs
Publication statusPublished - Mar 1 2018
Event21st International Conference on Principles of Distributed Systems, OPODIS 2017 - Lisboa, Portugal
Duration: Dec 18 2017Dec 20 2017

Other

Other21st International Conference on Principles of Distributed Systems, OPODIS 2017
CountryPortugal
CityLisboa
Period12/18/1712/20/17

Fingerprint

Chirality
Chemical activation
Mobile robots
Data storage equipment
Communication

All Science Journal Classification (ASJC) codes

  • Software

Cite this

Di Luna, G. A., Flocchini, P., Santoro, N., Viglietta, G., & Yamauchi, Y. (2018). Shape formation by programmable particles. In 21st International Conference on Principles of Distributed Systems, OPODIS 2017 (Vol. 95). [31] Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.OPODIS.2017.31

Shape formation by programmable particles. / Di Luna, Giuseppe A.; Flocchini, Paola; Santoro, Nicola; Viglietta, Giovanni; Yamauchi, Yukiko.

21st International Conference on Principles of Distributed Systems, OPODIS 2017. Vol. 95 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018. 31.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Di Luna, GA, Flocchini, P, Santoro, N, Viglietta, G & Yamauchi, Y 2018, Shape formation by programmable particles. in 21st International Conference on Principles of Distributed Systems, OPODIS 2017. vol. 95, 31, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 21st International Conference on Principles of Distributed Systems, OPODIS 2017, Lisboa, Portugal, 12/18/17. https://doi.org/10.4230/LIPIcs.OPODIS.2017.31
Di Luna GA, Flocchini P, Santoro N, Viglietta G, Yamauchi Y. Shape formation by programmable particles. In 21st International Conference on Principles of Distributed Systems, OPODIS 2017. Vol. 95. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. 2018. 31 https://doi.org/10.4230/LIPIcs.OPODIS.2017.31
Di Luna, Giuseppe A. ; Flocchini, Paola ; Santoro, Nicola ; Viglietta, Giovanni ; Yamauchi, Yukiko. / Shape formation by programmable particles. 21st International Conference on Principles of Distributed Systems, OPODIS 2017. Vol. 95 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2018.
@inproceedings{2fa2cd9d23ef400ca1b218dd1b82a8ff,
title = "Shape formation by programmable particles",
abstract = "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 (S0, SF), allows the particles to form the final shape SF (given in input) starting from the initial shape S0, unknown to the particles. The final configuration will be an appropriate scaled-up copy of SF 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(n2) rounds and moves: this number of moves is also asymptotically worst-case optimal.",
author = "{Di Luna}, {Giuseppe A.} and Paola Flocchini and Nicola Santoro and Giovanni Viglietta and Yukiko Yamauchi",
year = "2018",
month = "3",
day = "1",
doi = "10.4230/LIPIcs.OPODIS.2017.31",
language = "English",
volume = "95",
booktitle = "21st International Conference on Principles of Distributed Systems, OPODIS 2017",
publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

}

TY - GEN

T1 - Shape formation by programmable particles

AU - Di Luna, Giuseppe A.

AU - Flocchini, Paola

AU - Santoro, Nicola

AU - Viglietta, Giovanni

AU - Yamauchi, Yukiko

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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 (S0, SF), allows the particles to form the final shape SF (given in input) starting from the initial shape S0, unknown to the particles. The final configuration will be an appropriate scaled-up copy of SF 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(n2) rounds and moves: this number of moves is also asymptotically worst-case optimal.

AB - 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 (S0, SF), allows the particles to form the final shape SF (given in input) starting from the initial shape S0, unknown to the particles. The final configuration will be an appropriate scaled-up copy of SF 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(n2) rounds and moves: this number of moves is also asymptotically worst-case optimal.

UR - http://www.scopus.com/inward/record.url?scp=85045636750&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045636750&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.OPODIS.2017.31

DO - 10.4230/LIPIcs.OPODIS.2017.31

M3 - Conference contribution

AN - SCOPUS:85045636750

VL - 95

BT - 21st International Conference on Principles of Distributed Systems, OPODIS 2017

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

ER -