Algorithmica

pp 1–27 | Cite as

Optimal Staged Self-Assembly of General Shapes

  • Cameron Chalk
  • Eric Martinez
  • Robert Schweller
  • Luis Vega
  • Andrew Winslow
  • Tim Wylie
Article

Abstract

We analyze the number of tile types t, bins b, and stages necessary to assemble \(n \times n\) squares and scaled shapes in the staged tile assembly model. For \(n \times n\) squares, we prove \(\mathcal {O}\left( \frac{\log {n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}\right) \) stages suffice and \(\varOmega \left( \frac{\log {n} - tb - t\log t}{b^2}\right) \) are necessary for almost all n. For shapes S with Kolmogorov complexity K(S), we prove \(\mathcal {O}\left( \frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t}\right) \) stages suffice and \(\varOmega \left( \frac{K(S) - tb - t\log t}{b^2}\right) \) are necessary to assemble a scaled version of S, for almost all S. We obtain similarly tight bounds when the more powerful flexible glues are permitted.

Keywords

DNA computing Biocomputing Staging 2HAM Hierarchical 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Cameron Chalk
    • 1
  • Eric Martinez
    • 1
  • Robert Schweller
    • 1
  • Luis Vega
    • 1
  • Andrew Winslow
    • 1
  • Tim Wylie
    • 1
  1. 1.Department of Computer ScienceUniversity of Texas Rio Grande ValleyEdinburgUSA

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