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Optimal Program-Size Complexity for Self-Assembly at Temperature 1 in 3D

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9211)

Abstract

Working in a three-dimensional variant of Winfree’s abstract Tile Assembly Model, we show that, for all \(N \in \mathbb {N}\), there is a tile set that uniquely self-assembles into an \(N \times N\) square shape at temperature 1 with optimal program-size complexity of \(O(\log N / \log \log N)\) (the program-size complexity, also known as tile complexity, of a shape is the minimum number of unique tile types required to uniquely self-assemble it). Moreover, our construction is “just barely” 3D in the sense that it works even when the placement of tiles is restricted to the \(z = 0\) and \(z = 1\) planes. This result affirmatively answers an open question from Cook, Fu, Schweller (SODA 2011). To achieve this result, we develop a general 3D temperature 1 optimal encoding construction, reminiscent of the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree (SICOMP 2007), and perhaps of independent interest.

Keywords

  • Tile Complexity
  • Tile Type
  • Abstract Tile Assembly Model
  • Tile Set
  • Unique Tile Types

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    Technically, Rothemund and Winfree established the 2D self-assembly case, but their proof easily generalizes to 3D self-assembly.

  2. 2.

    One subtle difference between our 3D definition of K and the original 2D definition of the tile complexity of an \(N \times N\) square, given by Rothemund and Winfree in [7], is that they assume a fully-connected final structure, whereas we do not.

References

  1. Adleman, L., Cheng, Q., Goel, A., Huang, M.D.: Running time and program size for self-assembled squares. In: STOC, Huang, pp. 740–748 (2001)

    Google Scholar 

  2. Cook, M., Fu, Y., Schweller, R.: Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: SODA 2011: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM (2011)

    Google Scholar 

  3. Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature 1. Theor. Comput. Sci. 412, 145–158 (2011)

    MathSciNet  CrossRef  Google Scholar 

  4. Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and its Applications, 3rd edn. Springer, New York (2008)

    CrossRef  Google Scholar 

  5. Manuch, J., Stacho, L., Stoll, C.: Two lower bounds for self-assemblies at temperature 1. J. Comput. Biol. 17(6), 841–852 (2010)

    MathSciNet  CrossRef  Google Scholar 

  6. Presburger, Mojżesz: Über die vollständigkeit eines gewissen systems der arithmetik ganzer zahlen, welchem die addition als einzige operation hervortritt. In: Compte-Rendus du Premier Congrès des Mathématiciens des Pays Slaves, Warsaw, pp. 92–101 (1930)

    Google Scholar 

  7. Rothemund, P. W., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: STOC 2000: Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, pp. 459–468 (2000)

    Google Scholar 

  8. Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. 36(6), 1544–1569 (2007)

    MathSciNet  CrossRef  Google Scholar 

  9. Wang, H.: Proving theorems by pattern recognition - II. Bell Syst. Tech. J. XL 40(1), 1–41 (1961)

    CrossRef  Google Scholar 

  10. Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998

    Google Scholar 

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Correspondence to Scott M. Summers .

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Furcy, D., Micka, S., Summers, S.M. (2015). Optimal Program-Size Complexity for Self-Assembly at Temperature 1 in 3D. In: Phillips, A., Yin, P. (eds) DNA Computing and Molecular Programming. DNA 2015. Lecture Notes in Computer Science(), vol 9211. Springer, Cham. https://doi.org/10.1007/978-3-319-21999-8_5

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  • DOI: https://doi.org/10.1007/978-3-319-21999-8_5

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