Complexities for High-Temperature Two-Handed Tile Self-assembly

  • Robert Schweller
  • Andrew WinslowEmail author
  • Tim Wylie
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10467)


Tile self-assembly is a formal model of computation capturing DNA-based nanoscale systems. Here we consider the popular two-handed tile self-assembly model or 2HAM. Each 2HAM system includes a temperature parameter, which determines the threshold of bonding strength required for two assemblies to attach. Unlike most prior study, we consider general temperatures not limited to small, constant values. We obtain two results. First, we prove that the computational complexity of determining whether a given tile system uniquely assembles a given assembly is coNP-complete, confirming a conjecture of Cannon et al. (2013). Second, we prove that larger temperature values decrease the minimum number of tile types needed to assemble some shapes. In particular, for any temperature \(\tau \in \{3, \dots \}\), we give a class of shapes of size n such that the ratio of the minimum number of tiles needed to assemble these shapes at temperature \(\tau \) and any temperature less than \(\tau \) is \(\varOmega (n^{1/(2\tau +2)})\).


  1. 1.
    Abel, Z., Benbernou, N., Damian, M., Demaine, E.D., Demaine, M.L., Flatland, R., Kominers, S.D., Schweller, R.T.: Shape replication through self-assembly and RNase enzymes. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1045–1064 (2010)Google Scholar
  2. 2.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pp. 740–748 (2001)Google Scholar
  3. 3.
    Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D.A., Kempe, D., de Espanés, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pp. 23–32 (2002)Google Scholar
  4. 4.
    Cannon, S., Demaine, E.D., Demaine, E.D., Eisenstat, S., Patitz, M.J., Schweller, R., Summers, S.M., Winslow, A.: Two hands are better than one (up to constant factors): self-assembly in the 2HAM vs. aTAM. In: Proceedings of 30th International Symposium on Theoretical Aspects of Computer Science (STACS). LIPIcs, vol. 20, pp. 172–184. Schloss Dagstuhl (2013)Google Scholar
  5. 5.
    Chen, H.-L., Doty, D.: Parallelism and time in hierarchical self-assembly. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, 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.D., Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Woods, D.: The two-handed tile assembly model is not intrinsically universal. Algorithmica 74(2), 812–850 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Doty, D.: Producibility in hierarchical self-assembly. Nat. Comput. 15(1), 41–49 (2016)MathSciNetCrossRefzbMATHGoogle 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., Reishus, D., Schweller, R.T., Summers, S.M.: Strong fault-tolerance for self-assembly with fuzzy temperature. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 417–426 (2010)Google Scholar
  12. 12.
    Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature one. In: Deaton, R., Suyama, A. (eds.) DNA 2009. LNCS, vol. 5877, pp. 35–44. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-10604-0_4 CrossRefGoogle Scholar
  13. 13.
    Itai, A., Papadimitriou, C.H., Szwarcfiter, J.L.: Hamilton paths in grid graphs. SIAM J. Comput. 11(4), 676–686 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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, pp. 571–580 (2006)Google Scholar
  15. 15.
    Luhrs, C.: Polyomino-safe DNA self-assembly via block replacement. Nat. Comput. 9(1), 97–109 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Meunier, P.-E.: The self-assembly of paths and squares at temperature 1. Technical report, arXiv (2013).
  18. 18.
    Meunier, P.-E., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 752–771 (2014)Google Scholar
  19. 19.
    Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Winslow, A.: Resiliency to multiple nucleation in temperature-1 self-assembly. In: Rondelez, Y., Woods, D. (eds.) DNA 2016. LNCS, vol. 9818, pp. 98–113. Springer, Cham (2016). doi: 10.1007/978-3-319-43994-5_7 CrossRefGoogle Scholar
  20. 20.
    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
  21. 21.
    Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. SIAM J. Comput. 39(4), 1581–1616 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Schweller, R., Winslow, A., Wylie, T.: Verification in staged tile self-assembly. In: Patitz, M.J., Stannett, M. (eds.) UCNC 2017. LNCS, vol. 10240, pp. 98–112. Springer, Cham (2017). doi: 10.1007/978-3-319-58187-3_8 CrossRefGoogle Scholar
  23. 23.
    Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)CrossRefGoogle Scholar
  24. 24.
    Seki, S., Ukuno, Y.: On the behavior of tile assembly system at high temperatures. Computability 2(2), 107–124 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. 36(6), 1544–1569 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Summers, S.M.: Reducing tile complexity for the self-assembly of scaled shapes through temperature programming. Algorithmica 63(1), 117–136 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology (1998)Google Scholar
  28. 28.
    Woods, D.: Intrinsic universality and the computational power of self-assembly. Philos. Trans. R. Soc. A 373, 2015 (2046)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of Texas - Rio Grande ValleyEdinburgUSA

Personalised recommendations