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Natural Computing

Optimal staged self-assembly of linear assemblies

  • 06-05-2019
Published in:
Authors
Cameron Chalk
Eric Martinez
Robert Schweller
Luis Vega
Andrew Winslow
Tim Wylie
Published in
Natural Computing | Issue 3/2019

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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.

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previous article The alchemy of computation: designing with the unknown
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Title
Optimal staged self-assembly of linear assemblies
Authors
Cameron Chalk
Eric Martinez
Robert Schweller
Luis Vega
Andrew Winslow
Tim Wylie
Publication date
06-05-2019
Publisher
Springer Netherlands
Published in
Natural Computing / Issue 3/2019
Print ISSN: 1567-7818
Electronic ISSN: 1572-9796
DOI
https://doi.org/10.1007/s11047-019-09740-y
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