Theory of Distributed Systems

Universal Shape Formation

for Programmable Matter(Thim Strothmann)

Joint work with

BDA 2016 – July 25, 2016

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Theory of Distributed

Systems

Inspiration

BDA 2016

Video on this slide was deleted to decrease file size.

For inspiration Video (scene from Big Hero six) visit

https://www.youtube.com/watch?v=fF1rDKEC0TI.

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Theory of Distributed

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Motivation - Applications

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Theory of Distributed

Systems

The amoebot Model

BDA 2016

Overarching Constraint: Maintain Connectivity

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Theory of Distributed

Systems

The amoebot Model

BDA 2016

• “Standard” asynchronous computation model

• Only one particle is activated in each time step

• Once activated a particle can compute, communicate and perform one

move

• an adversary activates particles

• Round: every particle is activated at least once

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Theory of Distributed

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Shape Formation Problem

BDA 2016

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Theory of Distributed

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Naive Shape Formation Problem

BDA 2016

Video on this slide was deleted to decrease file size.

For naïve Shape Formation algorithms (Hexagon & Triangle) visit

http://sops.cs.upb.de/ .

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Theory of Distributed

Systems

Universal Shape Formation Problem

In general not possible, i.e.,

BDA 2016

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Theory of Distributed

Systems

Universal Shape Formation Problem

Input: constant size set of faces

Goal:

build shape given by faces (scaled-up and possibly rotated)

scale to include all particles

(no leftover particles)

BDA 2016

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Theory of Distributed

Systems

Universal Shape Formation Problem

BDA 2016

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Theory of Distributed

Systems

Universal Shape Formation Problem

Our Result:

Given any shape described by a constant number of faces, our algorithm

builds that shape using all particles in the system in 𝑂 𝑛 rounds.

BDA 2016

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Theory of Distributed

Systems

Universal Shape Formation Problem

Note: 𝑂 𝑛 rounds is not possible if we start in an arbitrary initial

configuration.

Solution:

BDA 2016

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Theory of Distributed

Systems

Universal Shape Formation Algorithm

Movement Primitives:

2) Triangle expansion/ contraction/ rotation (𝑂 ℓ rounds)

expansion contraction

rotation

ℓ

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Theory of Distributed

Systems

Universal Shape Formation Algorithm

Intermediate Structure

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Theory of Distributed

Systems

Universal Shape Formation Algorithm

Intermediate Structure

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Theory of Distributed

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Universal Shape Formation Algorithm

Building the final shape:

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Theory of Distributed

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Universal Shape Formation Algorithm

The devil is in the details:

Take care of the imperfection of the intermediate structure without

moving it.

Make up for the estimation errors of ℓ.

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Theory of Distributed

Systems

Universal Shape Formation Algorithm

The devil is in the details:

Triangles have to be cut to different sizes (+ incorporate waste).

The intermediate structure might block the final building process

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Theory of Distributed

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Summary & Future Work

Result: Given any shape described by a constant number of faces, our

algorithm builds that shape using all particles in the system in 𝑂 𝑛rounds.

Interesting Challenges:

arbitrary configuration of low diameter

non-constant size shapes

3D

Failures

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Theory of Distributed

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References

Corresponding Publication:Zahra Derakhshandeh, Robert Gmyr, Andréa W. Richa, Christian Scheideler, Thim Strothmann:

Universal Shape Formation for Programmable Matter. SPAA 2016: 289-299

(http://doi.acm.org/10.1145/2935764.2935784)

For videos of some of our algorithms and a (slightly outdated)

publication history in the topic visit:http://sops.cs.upb.de/

BDA 2016