Skip to main content

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)


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.


  • 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.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-21999-8_5
  • Chapter length: 16 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
USD   54.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-21999-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   69.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.


  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.


  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 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Scott M. Summers .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

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.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21998-1

  • Online ISBN: 978-3-319-21999-8

  • eBook Packages: Computer ScienceComputer Science (R0)