Optimal Staged Self-Assembly of General Shapes

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.

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Notes

  1. 1.

    Such a restriction is necessary, as systems with a single tile type are incapable of assembling finite non-trivial shapes.

  2. 2.

    The fraction of values for which the statement holds reaches 1 in the limit as \(n \rightarrow \infty \).

  3. 3.

    For more information on Kolmogorov complexity, we suggest [19].

  4. 4.

    The original staged model [8] only considered \(\mathcal {O}\left( 1\right) \) distinct tile types, and thus for simplicity allowed tiles to be added at any stage (since \(\mathcal {O}(1)\) extra bins could hold the individual tile types to mix at any stage). Because systems here may have super-constant tile complexity, we restrict tiles to only be added at the initial stage.

  5. 5.

    This is a slight modification of the original staged model [8] in that there is no requirement of a final stage with a single output bin. It may be easier in general to solve problems in this variant of the model, so it is considered for lower bound purposes. However, all results herein apply to both variants of the model.

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Correspondence to Andrew Winslow.

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Research supported in part by National Science Foundation Grants CCF-1117672, CCF-1555626, and CCF-1422152.

Andrew Winslow was previously at Université Libre de Bruxelles.

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Chalk, C., Martinez, E., Schweller, R. et al. Optimal Staged Self-Assembly of General Shapes. Algorithmica 80, 1383–1409 (2018). https://doi.org/10.1007/s00453-017-0318-0

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Keywords

  • DNA computing
  • Biocomputing
  • Staging
  • 2HAM
  • Hierarchical