Particle-Based Assembly Using Precise Global Control

TL;DR

This paper investigates the computational complexity of assembling shapes from particles under global control, proving NP-completeness in 3D, and providing algorithms for constructibility decisions and shape scaling effects.

Contribution

It introduces the single step model for particle assembly, proves NP-completeness for 3D shape constructibility, and offers algorithms for shape decision and scaling properties.

Findings

Deciding 3D shape constructibility is NP-complete.

A maximum constructible shape can be approximated.

Scaling shapes ensures constructibility, e.g., 2-scaled polyominoes and 3-scaled polycubes are constructible.

Abstract

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., a model in which each particle moves one unit step into the given direction. We restrict the assembly process such that at each single time step actually one particle is added to and moved within the workspace. 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.