Optimal staged self-assembly of linear assemblies

Abstract

We analyze the complexity of building linear assemblies, sets of linear assemblies, and ${\mathcal{O}}(1)$-scale general shapes in the staged tile assembly model. For systems with at most b bins and t tile types, we prove that the minimum number of stages to uniquely assemble a $1 \times n$line is $\varTheta (\log _t{n} + \log _b{\frac{n}{t}} + 1)$. Generalizing to ${\mathcal{O}}(1) \times n$ lines, we prove the minimum number of stages is ${\mathcal{O}}(\frac{\log {n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ and $\varOmega (\frac{\log {n} - tb - t\log t}{b^2})$. We also obtain similar upper and lower bounds in a model permitting flexible glues using non-diagonal glue functions. Next, we consider assembling sets of lines and general shapes using $t = {\mathcal{O}}(1)$ tile types. We prove that the minimum number of stages needed to assemble a set of k lines of size at most ${\mathcal{O}}(1) \times n$ is ${\mathcal{O}}(\frac{k\log n}{b^2}+\frac{k\sqrt{\log n}}{b}+\log \log n)$ and $\varOmega (\frac{k\log n}{b^2})$. In the case that $b = \mathcal {O}(\sqrt{k})$, the minimum number of stages is $\varTheta (\log {n})$. The upper bound in this special case is then used to assemble “hefty” shapes of at least logarithmic edge-length-to-edge-count ratio at $\mathcal {O}(1)$-scale using $\mathcal {O}(\sqrt{k})$ bins and optimal $\mathcal {O}(\log {n})$ stages.