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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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Copyright information

© 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

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