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 \(\varOmega (\frac{\log {n} - tb - t\log t}{b^2})\).