License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2017.11
URN: urn:nbn:de:0030-drops-82214
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/8221/
Becker, Aaron T. ;
Fekete, Sándor P. ;
Keldenich, Phillip ;
Krupke, Dominik ;
Rieck, Christian ;
Scheffer, Christian ;
Schmidt, Arne
Tilt Assembly: Algorithms for Micro-Factories that Build Objects with Uniform External Forces
Abstract
We present algorithmic results for the parallel assembly of many micro-scale objects in two and three dimensions from tiny particles, which has been proposed in the context of programmable matter and self-assembly for building high-yield micro-factories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle; particles can bond when being forced together with another appropriate particle.
Due to the physical and geometric constraints, not all shapes can be built in this manner; this gives rise to the Tilt Assembly Problem (TAP) of deciding constructibility. For simply-connected polyominoes P in 2D consisting of N unit-squares ("tiles"), we prove that TAP can be decided in O(N log N) time. For the optimization variant MaxTAP (in which the
objective is to construct a subshape of maximum possible size), we show polyAPX-hardness: unless P=NP, MaxTAP cannot be approximated within a factor of N^(1/3); for tree-shaped structures, we give an N^(1/2)-approximation algorithm. For the efficiency of the assembly process itself, we show that any constructible shape allows pipelined assembly, which produces copies of P in O(1) amortized time, i.e., N copies of P in O(N) time steps. These considerations can be extended to three-dimensional objects: For the class of polycubes P we prove that it is NP-hard to decide whether it is possible to construct a path between two points of P; it is also NP-hard to decide constructibility of a polycube P. Moreover, it is expAPX-hard to maximize a path from a given start point.
BibTeX - Entry
@InProceedings{becker_et_al:LIPIcs:2017:8221,
author = {Aaron T. Becker and S{\'a}ndor P. Fekete and Phillip Keldenich and Dominik Krupke and Christian Rieck and Christian Scheffer and Arne Schmidt},
title = {{Tilt Assembly: Algorithms for Micro-Factories that Build Objects with Uniform External Forces}},
booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)},
pages = {11:1--11:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-054-5},
ISSN = {1868-8969},
year = {2017},
volume = {92},
editor = {Yoshio Okamoto and Takeshi Tokuyama},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/8221},
URN = {urn:nbn:de:0030-drops-82214},
doi = {10.4230/LIPIcs.ISAAC.2017.11},
annote = {Keywords: Programmable matter, micro-factories, tile assembly, tilt, approximation, hardness}
}
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Keywords: |
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Programmable matter, micro-factories, tile assembly, tilt, approximation, hardness |
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Collection: |
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28th International Symposium on Algorithms and Computation (ISAAC 2017) |
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Issue Date: |
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2017 |
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Date of publication: |
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07.12.2017 |
