Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Furcy, David; Summers, Scott M.; Withers, Logan https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-146716
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Improved Lower and Upper Bounds on the Tile Complexity of Uniquely Self-Assembling a Thin Rectangle Non-Cooperatively in 3D

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Abstract

We investigate a fundamental question regarding a benchmark class of shapes in one of the simplest, yet most widely utilized abstract models of algorithmic tile self-assembly. More specifically, we study the directed tile complexity of a k × N thin rectangle in Winfree’s ubiquitous abstract Tile Assembly Model, assuming that cooperative binding cannot be enforced (temperature-1 self-assembly) and that tiles are allowed to be placed at most one step into the third dimension (just-barely 3D). While the directed tile complexities of a square and a scaled-up version of any algorithmically specified shape at temperature 1 in just-barely 3D are both asymptotically the same as they are (respectively) at temperature 2 in 2D, the (nearly tight) bounds on the directed tile complexity of a thin rectangle at temperature 2 in 2D are not currently known to hold at temperature 1 in just-barely 3D. Motivated by this discrepancy, we establish new lower and upper bounds on the directed tile complexity of a thin rectangle at temperature 1 in just-barely 3D. The proof of our upper bound is based on the construction of a novel, just-barely 3D temperature-1 self-assembling counter. Each value of the counter is comprised of k-2 digits, represented in a geometrically staggered fashion within k rows. This nearly optimal digit density, along with the base of the counter, which is proportional to N^{1/(k-1)}, results in an upper bound of O(N^{1/(k-1)} + log N), and is an asymptotic improvement over the previous state-of-the-art upper bound. On our way to proving our lower bound, we develop a new, more powerful type of specialized Window Movie Lemma that lets us bound the number of "sufficiently similar" ways to assign glues to a set (rather than a sequence) of fixed locations. Consequently, our lower bound, Ω(N^{1/k}), is also an asymptotic improvement over the previous state-of-the-art lower bound.

BibTeX - Entry

@InProceedings{furcy_et_al:LIPIcs.DNA.27.4,
  author =	{Furcy, David and Summers, Scott M. and Withers, Logan},
  title =	{{Improved Lower and Upper Bounds on the Tile Complexity of Uniquely Self-Assembling a Thin Rectangle Non-Cooperatively in 3D}},
  booktitle =	{27th International Conference on DNA Computing and Molecular Programming (DNA 27)},
  pages =	{4:1--4:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-205-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{205},
  editor =	{Lakin, Matthew R. and \v{S}ulc, Petr},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14671},
  URN =		{urn:nbn:de:0030-drops-146716},
  doi =		{10.4230/LIPIcs.DNA.27.4},
  annote =	{Keywords: Self-assembly, algorithmic self-assembly, tile self-assembly}
}

Keywords: Self-assembly, algorithmic self-assembly, tile self-assembly
Seminar: 27th International Conference on DNA Computing and Molecular Programming (DNA 27)
Issue date: 2021
Date of publication: 08.09.2021


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