Tilt Assembly: Algorithms for Micro-Factories That Build Objects with Uniform External Forces


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(NlogN) 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 Ω(N13); for tree-shaped structures, we give an O(N12)-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.



Programmable matter, Micro-factories, Tile assembly, Tilt, Approximation, Hardness


Copyright 2018 Algorithmica. This is a post-print version of a published paper that is available at: https://link.springer.com/article/10.1007/s00453-018-0483-9 Recommended citation: Becker, Aaron T., Sándor P. Fekete, Phillip Keldenich, Dominik Krupke, Christian Rieck, Christian Scheffer, and Arne Schmidt. "Tilt assembly: algorithms for micro-factories that build objects with uniform external forces." Algorithmica (2017): 1-23. DOI: 10.1007/s00453-018-0483-9 This item has been deposited in accordance with publisher copyright and licensing terms and with author's permission.