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.
@InProceedings{becker_et_al:LIPIcs.ISAAC.2017.11, author = {Becker, Aaron T. and Fekete, S\'{a}ndor P. and Keldenich, Phillip and Krupke, Dominik and Rieck, Christian and Scheffer, Christian and Schmidt, Arne}, 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 = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.11}, 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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