Skip to main content
SpringerLink
Log in

Reducing Tile Complexity for the Self-assembly of Scaled Shapes Through Temperature Programming

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

This paper concerns the self-assembly of scaled-up versions of arbitrary finite shapes. We work in the multiple temperature model that was introduced by Aggarwal, Cheng, Goldwasser, Kao, and Schweller (Complexities for Generalized Models of Self-Assembly, SIAM J. Comput. 2005). The multiple temperature model is a natural generalization of Winfree’s abstract tile assembly model, where the temperature of a tile system is allowed to be shifted up and down as self-assembly proceeds. We first exhibit two constant-size tile sets in which scaled-up versions of arbitrary shapes self-assemble. Our first tile set has the property that each scaled shape self-assembles via an asymptotically “Kolmogorov-optimum” temperature sequence but the scaling factor grows with the size of the shape being assembled. In contrast, our second tile set assembles each scaled shape via a temperature sequence whose length is proportional to the number of points in the shape but the scaling factor is a constant independent of the shape being assembled. We then show that there is no constant-size tile set that can uniquely assemble an arbitrary (non-scaled, connected) shape in the multiple temperature model, i.e., the scaling is necessary for self-assembly. This answers an open question of Kao and Schweller (Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 571–580, 2006), who asked whether such a tile set exists.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adleman, L., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: Proceedings of the Thirty-third Annual ACM Symposium on Theory of Computing (STOC 2001), New York, NY, USA, pp. 740–748. ACM, New York (2001)

    Chapter  Google Scholar 

  2. Barish, R.D., Schulman, R., Rothemund, P.W., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proc. Natl. Acad. Sci. USA 106(15), 6054–6059 (2009)

    Article  Google Scholar 

  3. Becker, F., Rapaport, I., Rémila, E.: Self-assembling classes of shapes with a minimum number of tiles, and in optimal time. In: Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp. 45–56 (2006)

    Google Scholar 

  4. Chen, H.-L., Goel, A.: Error free self-assembly with error prone tiles. In: Proceedings of the 10th International Meeting on DNA Based Computers (2004)

    Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Demaine, E.D., Demaine, M.L., Fekete, S.P., Ishaque, M., Rafalin, E., Schweller, R.T., Souvaine, D.L.: Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues. Nat. Comput. 7(3), 347–370 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Doty, D.: Randomized self-assembly for exact shapes. SIAM J. Comput. 39(8), 3521–3552 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Doty, D., Lutz, J.H., Patitz, M.J., Summers, S.M., Woods, D.: Intrinsic universality in self-assembly. In: Marion, J.-Y., Schwentick, T. (eds.) Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science (STACS 2010), Dagstuhl, Germany. Leibniz International Proceedings in Informatics (LIPIcs), vol. 5, pp. 275–286. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, Wadern (2010)

    Google Scholar 

  9. Kao, M.-Y., Schweller, R.T.: Reducing tile complexity for self-assembly through temperature programming. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), Miami, Florida, Jan. 2006, pp. 571–580 (2007)

    Google Scholar 

  10. Kao, M.-Y., Schweller, R.T.: Randomized self-assembly for approximate shapes. In: International Colloqium on Automata, Languages, and Programming (ICALP). Lecture Notes in Computer Science, vol. 5125, pp. 370–384. Springer, Berlin (2008)

    Chapter  Google Scholar 

  11. Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. Theor. Comput. Sci. 410, 384–405 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Reif, J.H.: Local parallel biomolecular computing. In: DNA Based Computers III. DIMACS, vol. 48, pp. 217–254. Am. Math. Soc., Providence (1999)

    Google Scholar 

  13. Rothemund, P.W.K.: Theory and experiments in algorithmic self-assembly. Ph.D. thesis, University of Southern California, December 2001

  14. Rothemund, P.W.K.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006)

    Article  Google Scholar 

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

    Chapter  Google Scholar 

  16. Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2(12), 2041–2053 (2004)

    Article  Google Scholar 

  17. Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Vitányi, P., Li, M.: An Introduction to Kolmogorov Complexity and Its Applications. Springer, Berlin (1997)

    MATH  Google Scholar 

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

    Google Scholar 

  21. Wang, H.: Dominoes and the AEA case of the decision problem. In: Proceedings of the Symposium on Mathematical Theory of Automata, New York, 1962, pp. 23–55. Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn (1963)

    Google Scholar 

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

  23. Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. In: Chen, J., Reif, J.H. (eds.) DNA. Lecture Notes in Computer Science, vol. 2943, pp. 126–144. Springer, Berlin (2003)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Software Engineering, University of Wisconsin–Platteville, Platteville, WI, 53818, USA

    Scott M. Summers

Authors
  1. Scott M. Summers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Scott M. Summers.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Summers, S.M. Reducing Tile Complexity for the Self-assembly of Scaled Shapes Through Temperature Programming. Algorithmica 63, 117–136 (2012). https://doi.org/10.1007/s00453-011-9522-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-011-9522-5

Keywords

Search

Navigation