Particle-Based Assembly Using Precise Global Control

In micro- and nano-scale systems, particles can be moved by using an external force like gravity or a magnetic field. In the presence of adhesive particles that can attach to each other, the challenge is to decide whether a shape is constructible. Previous work provides a class of shapes for which constructibility can be decided efficiently, when particles move maximally into the same direction induced by a global signal. In this paper we consider the single step model, i.e., each particle moves one unit step into the given direction. We prove that deciding constructibility is NP-complete for three-dimensional shapes, and that a maximum constructible shape can be approximated. The same approximation algorithm applies for 2D. We further present linear-time algorithms to decide whether or not a tree-shape in 2D or 3D is constructible. Scaling a shape yields constructibility; in particular we show that the $2$-scaled copy of every non-degenerate polyomino is constructible. In the three-dimensional setting we show that the $3$-scaled copy of every non-degenerate polycube is constructible.