Browsing by Author "Schmidt, Arne"
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Item Efficient Parallel Self- Assembly Under Uniform Control Inputs(IEEE Robotics and Automation Letters, 7/9/2018) Becker, Aaron T.; Schmidt, Arne; Manzoor, Sheryl; Huang, Li; Fekete, Sándor P.We prove that by successively combining subassemblies, we can achieve sublinear construction times for “staged” assembly of microscale objects from a large number of tiny particles, for vast classes of shapes; this is a significant advance in the context of programmable matter and self-assembly for building high-yield microfactories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle; particles bond when forced together with a compatible particle. Previous work considered sequential composition of objects, resulting in construction time that is linear in the number N of particles, which is inefficient for large N. Our progress implies critical speedup for constructible shapes; for convex polyominoes, even a constant construction time is possible. We also show that our construction process can be used for pipelining, resulting in an amortized constant production time.Item Tilt Assembly: Algorithms for Micro-Factories That Build Objects with Uniform External Forces(Algorithmica, 8/3/2018) Becker, Aaron T.; Fekete, Sándor P.; Keldenich, Phillip; Krupke, Dominik; Rieck, Christian; Scheffer, Christian; Schmidt, ArneWe 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 $\Omega(N^{\frac{1}{3}})$; for tree-shaped structures, we give an $O(N^{\frac{1}{2}})$-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.