Optimal Staged Self-Assembly of General Shapes

TL;DR

This paper establishes tight bounds on the number of stages needed for self-assembling arbitrary shapes and squares using tile assembly models, considering parameters like tile types, bins, and stages, with implications for efficient shape construction.

Contribution

It provides the first tight bounds on staged self-assembly complexity for general shapes and squares, incorporating parameters like tile types, bins, and stages, and extends results to flexible glue models.

Findings

Stages depend on shape complexity and assembly parameters.

Bounds are tight for almost all shapes and sizes.

Flexible glues do not significantly reduce assembly stages.

Abstract

We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \times n$ squares and scaled shapes in the staged tile assembly model. For $n \times n$ squares, we prove $\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{\log{n} - tb - t\log t}{b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{K(S) - tb - t\log t}{b^2})$ are necessary to assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight bounds when the more powerful flexible glues are permitted.