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Algorithmica

pp 1–22 | Cite as

Nearly Constant Tile Complexity for any Shape in Two-Handed Tile Assembly

  • Robert Schweller
  • Andrew Winslow
  • Tim WylieEmail author
Article
  • 13 Downloads

Abstract

Tile self-assembly is a well-studied theoretical model of geometric computation based on nanoscale DNA-based molecular systems. Here, we study the two-handed tile self-assembly model or 2HAM at general temperatures, in contrast with prior study limited to small constant temperatures, leading to surprising results. We obtain constructions at larger (i.e., hotter) temperatures that disprove prior conjectures and break well-known bounds for low-temperature systems via new methods of temperature-encoded information. In particular, for all \(n \in \mathbb {N}\), we assemble \(n \times n\) squares using \(O(2^{\log ^*{n}})\) tile types, thus breaking the well-known information theoretic lower bound of Rothemund and Winfree. Using this construction, we then show how to use the temperature to encode general shapes and construct them at scale with \(O(2^{\log ^*{K}})\) tiles, where K denotes the Kolmogorov complexity of the target shape. Following, we refute a long-held conjecture by showing how to use temperature to construct \(n \times O(1)\) rectangles using only \(O(\log {n}/\log \log {n})\) tile types. We also give two small systems to generate nanorulers of varying length based solely on varying the system temperature. These results constitute the first real demonstration of the power of high temperature systems for tile assembly in the 2HAM. This leads to several directions for future explorations which we discuss in the conclusion.

Keywords

Self-assembly Hierarchical assembly 2HAM 

Notes

References

  1. 1.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.D.: Running time and program size for self-assembled squares. In: Proceedings of the 33rd ACM Symposium on Theory of Computing, STOC’01, pp. 740–748 (2001)Google Scholar
  2. 2.
    Bryans, N., Chiniforooshan, E., Doty, D., Kari, L., Seki, S.: The power of nondeterminism in self-assembly. In: Proceedings of the 22nd ACM–SIAM Symposium on Discrete Algorithms, SODA’11, pp. 590–602. SIAM (2011)Google Scholar
  3. 3.
    Cannon, S., Demaine, E.D., Demaine, M.L., Eisenstat, S., Patitz, M.J., Schweller, R., Summers, S.M., Winslow, A.: Two hands are better than one (up to constant factors). In: Proceedings of 30th International Symposium on Theory Aspects of Computing Science, STACS’13, pp. 172–184 (2013)Google Scholar
  4. 4.
    Chalk, C., Luchsinger, A., Schweller, R., Wylie, T.: Self-assembly of any shape with constant tile types using high temperature. In: Proceedings of the 26th Annual European Symposium on Algorithms, ESA’18 (2018)Google Scholar
  5. 5.
    Chen, H.L., Doty, D.: Parallelism and time in hierarchical self-assembly. In: Proceedings of the 23rd ACM–SIAM Symposium on Discrete Algorithms, SODA’12, pp. 1163–1182. SIAM (2012)Google Scholar
  6. 6.
    Chen, H.L., Doty, D., Seki, S.: Program size and temperature in self-assembly. Algorithmica 72(3), 884–899 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheng, Q., Aggarwal, G., Goldwasser, M.H., Kao, M.Y., Schweller, R.T., de Espanés, P.M.: Complexities for generalized models of self-assembly. SIAM J. Comput. 34, 1493–1515 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Demaine, E., Patitz, M., Rogers, T., Schweller, R., Summers, S.M., Woods, D.: The two-handed tile assembly model is not intrinsically universal. In: Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP 2013) (2013)Google Scholar
  9. 9.
    Doty, D.: Personal communicationGoogle Scholar
  10. 10.
    Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: Proceedings of the 53rd IEEE Conference on Foundations of Computer Science (FOCS), pp. 302–310 (2012)Google Scholar
  11. 11.
    Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature one. In: Proceedings of 15th International Conference on DNA Computing and Molecular Programming (DNA’09), LNCS, vol. 5877, pp. 35–44. Springer, Berlin (2009)Google Scholar
  12. 12.
    Evans, C.: Crystals that count! Physical principles and experimental investigations of DNA tile self-assembly. Ph.D. thesis, California Institute of Technology (2014)Google Scholar
  13. 13.
    Maňuch, J., Stacho, L., Stoll, C.: Two lower bounds for self-assemblies at temperature 1. J. Comput. Biol. 16(6), 841–852 (2010)MathSciNetGoogle Scholar
  14. 14.
    Meunier, P.E.: The self-assembly of paths and squares at temperature 1. Tech. rep., arXiv (2013)Google Scholar
  15. 15.
    Meunier, P.E., Patitz, M.J., Summers, S.M., Theyssier, G., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: Proceedings of the 25th Symposium on Discrete Algorithms (SODA), pp. 752–771 (2014)Google Scholar
  16. 16.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC), pp. 459–468 (2000)Google Scholar
  17. 17.
    Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. SIAM J. Comput. 39(4), 1581–1616 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schweller, R.T., Winslow, A., Wylie, T.: Complexities for high-temperature two-handed tile self-assembly. In: 23rd International Conference on Computing and Molecular Programming, DNA’17, pp. 98–109 (2017).  https://doi.org/10.1007/978-3-319-66799-7_7
  19. 19.
    Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)CrossRefGoogle Scholar
  20. 20.
    Seki, S., Ukuno, Y.: On the behavior of tile assembly system at high temperatures. Computability 2(2), 107–124 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. 36(6), 1544–1569 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Summers, S.M.: Reducing tile complexity for the self-assembly of scaled shapes through temperature programming. Algorithmica 63(1), 117–136 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Winfree, E.: Algorithmic Self-Assembly of DNA. Ph.D. thesis, California Institute of Technology (1998)Google Scholar
  24. 24.
    Woods, D.: Intrinsic universality and the computational power of self-assembly. Philos. Trans. R. Soc. A (2015).  https://doi.org/10.1098/rsta.2014.0214

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of Texas - Rio Grande ValleyEdinburgUSA

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