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DOI: 10.4230/LIPIcs.ESA.2016.26

Optimal Staged Self-Assembly of General Shapes

Authors: Cameron Chalk, Eric Martinez, Robert Schweller, Luis Vega, Andrew Winslow, and Tim Wylie

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract

We analyze the number of stages, tiles, and bins needed to construct n * n squares and scaled shapes in the staged tile assembly model. In particular, we prove that there exists a staged system with b bins and t tile types assembling an n * n square using O((log n - tb - t log t)/b^2 + log log b/log t) stages and Omega((log n - tb - t log t)/b^2) are necessary for almost all n. For a shape S, we prove O((K(S) - tb - t log t)/b^2 + (log log b)/log t) stages suffice and Omega((K(S) - tb - t log t)/b^2) are necessary for the assembly of a scaled version of S, where K(S) denotes the Kolmogorov complexity of S. Similarly tight bounds are also obtained when more powerful flexible glue functions are permitted. These are the first staged results that hold for all choices of b and t and generalize prior results. The upper bound constructions use a new technique for efficiently converting each both sources of system complexity, namely the tile types and mixing graph, into a "bit string" assembly.

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Cameron Chalk, Eric Martinez, Robert Schweller, Luis Vega, Andrew Winslow, and Tim Wylie. Optimal Staged Self-Assembly of General Shapes. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chalk_et_al:LIPIcs.ESA.2016.26,
  author =	{Chalk, Cameron and Martinez, Eric and Schweller, Robert and Vega, Luis and Winslow, Andrew and Wylie, Tim},
  title =	{{Optimal Staged Self-Assembly of General Shapes}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{26:1--26:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.26},
  URN =		{urn:nbn:de:0030-drops-63776},
  doi =		{10.4230/LIPIcs.ESA.2016.26},
  annote =	{Keywords: Tile self-assembly, 2HAM, aTAM, DNA computing, biocomputing}
}

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Optimal Staged Self-Assembly of General Shapes

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