# Optimal Staged Self-assembly of Linear Assemblies

• Cameron Chalk
• Eric Martinez
• Robert Schweller
• Luis Vega
• Andrew Winslow
• Tim Wylie
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10867)

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

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.

## Keywords

Tile self-assembly Staged self-assembly DNA computing Biocomputing

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Cameron Chalk
• 1
• Eric Martinez
• 2
• Robert Schweller
• 2
• Luis Vega
• 2
• Andrew Winslow
• 2
• Tim Wylie
• 2
1. 1.Department of Electrical and Computer EngineeringUniversity of Texas at AustinAustinUSA
2. 2.Department of Computer ScienceUniversity of Texas - Rio Grande ValleyEdinburgUSA