Shape Formation by Programmable Particles Brief Announcement DISC - - PowerPoint PPT Presentation

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Shape Formation by Programmable Particles Brief Announcement DISC 2017 Giuseppe Antonio Di Luna, Paola Flocchini, Nicola Santoro, Giovanni Viglietta, Yukiko Yamauchi Vienna October 19, 2017 Shape Formation by Programmable Particles

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Shape Formation by Programmable Particles

Brief Announcement – DISC 2017 Giuseppe Antonio Di Luna, Paola Flocchini, Nicola Santoro, Giovanni Viglietta, Yukiko Yamauchi Vienna – October 19, 2017

Shape Formation by Programmable Particles

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Amoebots

/In this model, particles occupy nodes of a hexagonal grid./

Shape Formation by Programmable Particles

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Amoebots

/A particle can move by expanding and contracting./

Shape Formation by Programmable Particles

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Amoebots

/A particle can move by expanding and contracting./

Shape Formation by Programmable Particles

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Amoebots

/A particle can move by expanding and contracting./

Shape Formation by Programmable Particles

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Amoebots

/A particle can move by expanding and contracting./

Shape Formation by Programmable Particles

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Amoebots

/A particle can move by expanding and contracting./

Shape Formation by Programmable Particles

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Amoebots

/A system of particles is given./

Shape Formation by Programmable Particles

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Amoebots

/Particles move asynchronously./

Shape Formation by Programmable Particles

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Amoebots

/Particles move asynchronously./

Shape Formation by Programmable Particles

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Amoebots

/At each step, any set of particles is activated by an adversary./

Shape Formation by Programmable Particles

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Amoebots

/At each step, any set of particles is activated by an adversary./

Shape Formation by Programmable Particles

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Shape Formation

final shape

/The goal is to form a shape that is given as input to all particles./

Shape Formation by Programmable Particles

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Shape Formation

initial configuration final configuration deterministic algorithm

/The shape formation algorithm should be deterministic./

Shape Formation by Programmable Particles

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Shape Formation

initial configuration final configuration deterministic algorithm

/The shape can be scaled up depending on the size of the system./

Shape Formation by Programmable Particles

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Related Literature

Original Amoebot paper:

Z. Derakhshandeh, R. Gmyr, T. Strothmann, R.A. Bazzi,

A.W. Richa, and C. Scheideler Leader election and shape formation with self-organizing programmable matter In Proceedings of 21st International Conference on DNA Computing and Molecular Programming (DNA), 117–132, 2015

Randomized shape-formation algorithm for sequentially activated Amoebots starting from a triangular shape:

Z. Derakhshandeh, R. Gmyr, A.W. Richa, C. Scheideler, and
T. Strothmann

Universal shape formation for programmable matter In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 289–299, 2016

Shape Formation by Programmable Particles

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Our Particle Model

The n particles in the system: initially form any simply connected shape know the final shape but do not know n have a constant amount of internal memory are anonymous and start in the same state can only see and communicate with adjacent particles do not have a compass may not agree on a clockwise direction are activated asynchronously execute the same deterministic algorithm cannot occupy the same node

Shape Formation by Programmable Particles

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Unbreakable Symmetry

/If the system has a center of symmetry not in a grid node.../

Shape Formation by Programmable Particles

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Unbreakable Symmetry

/Then this symmetry is impossible to break./

Shape Formation by Programmable Particles

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Unbreakable Symmetry

/The same holds for systems with a 3-fold rotational symmetry./

Shape Formation by Programmable Particles

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Unbreakable Symmetry

/If the center is not in a grid node, the symmetry is unbreakable./

Shape Formation by Programmable Particles

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Statement of Results

Theorem If the system initially has an unbreakable symmetry, it cannot form shapes that do not have the same symmetry. Theorem For all other cases, there is a universal shape-formation algorithm, provided that the system is sufficiently large. The particles perform O(n2) moves in total and terminate in O(n2) rounds.

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Start with a sufficiently large simply connected system./

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Phase 1: attempt to elect a leader./

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Phase 2: construct a spanning forest./

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Phase 3: agree on a clockwise direction./

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Phase 4: form one line per leader./

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Phase 5: determine the scale of the final shape and assign roles./

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Phase 6: form the final shape./

Shape Formation by Programmable Particles

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Universal Shape-Formation Algorithm

/Phase 6: form the final shape./

Shape Formation by Programmable Particles

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Matching Lower Bound

/This example shows that O(n2) total moves are optimal./

Shape Formation by Programmable Particles

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Matching Lower Bound

/Open problem: are O(n2) rounds optimal?/

Shape Formation by Programmable Particles