New Geometric Algorithms for Fully Connected Staged Self-Assembly

  • Erik D. Demaine
  • Sándor P. Fekete
  • Christian Scheffer
  • Arne Schmidt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9211)

Abstract

We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et al. showed that a relatively small number of tile types suffices to produce arbitrary shapes in this model. However, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph in the case of shapes with holes; designing fully connected assemblies with a polylogarithmic number of stages was left as a major open problem. We resolve this challenge by presenting new systems for staged assembly that produce fully connected polyominoes in \(\mathcal {O}(\log ^2 n)\) stages, for various scale factors and temperature \(\tau =2\) as well as \(\tau =1\). Our constructions work even for shapes with holes and uses only a constant number of glues and tiles. Moreover, the underlying approach is more geometric in nature, implying that it promised to be more feasible for shapes with compact geometric description.

References

  1. 1.
    Abel, Z., Benbernou, N., Damian, M., Demaine, E.D., Demaine, M.L., Flatland, R., Kominers, S.D., Schweller, R.: Shape replication through self-assembly and rnase enzymes. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1045–1064 (2010)Google Scholar
  2. 2.
    Aggarwal, G., Cheng, Q., Goldwasser, M.H., Kao, M.-Y., de Espanes, P.M., Schweller, R.T.: Complexities for generalized models of self-assembly. SIAM J. Comput. 34(6), 1493–1515 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cannon, S., Demaine, E.D., Demaine, M.L., Eisenstat, S., Patitz, M.J., Schweller, R.T., Summers, S.M., Winslow, A.:. Two hands are better than one (up to constant factors). In: Symposium on Theoretical Aspects of Computer Science (STACS), pp. 172–184 (2013)Google Scholar
  4. 4.
    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. Natural Comput. 7(3), 347–370 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Demaine, E.D., Demaine, M.L., Fekete, S.P., Patitz, M.J., Schweller, R.T., Winslow, A., Woods, D.: One tile to rule them all: simulating any tile assembly system with a single universal tile. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 368–379. Springer, Heidelberg (2014) Google Scholar
  6. 6.
    Fekete, S.P., Hendricks, J., Patitz, M.J., Rogers, T.A., Schweller, R.T.: Universal computation with arbitrary polyomino tiles in non-cooperative self-assembly. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 148–167 (2015)Google Scholar
  7. 7.
    Fu, B., Patitz, M.J., Schweller, R.T., Sheline, R.: Self-assembly with geometric tiles. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 714–725. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  8. 8.
    Padilla, J.E., Liu, W., Seeman, N.C.: Hierarchical self assembly of patterns from the robinson tilings: DNA tile design in an enhanced tile assembly model. Natural Comput. 11(2), 323–338 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Padilla, J.E., Patitz, M.J., Schweller, R.T., Seeman, N.C., Summers, S.M., Zhong, X.: Asynchronous signal passing for tile self-assembly: fuel efficient computation and efficient assembly of shapes. Int. J. Found. Comput. Sci. 25(4), 459–488 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Park, S.H., Pistol, C., Ahn, S.J., Reif, J.H., Lebeck, A.R., Dwyer, C., LaBean, T.H.: Finite-size, fully addressable DNA tile lattices formed by hierarchical assembly procedures. Angewandte Chemie 118(5), 749–753 (2006)CrossRefGoogle Scholar
  11. 11.
    Reif, J.H.: Local parallel biomolecular computation. DNA-Based Comput. 3, 217–254 (1999)Google Scholar
  12. 12.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: ACM Symposium on Theory of Computing (STOC), pp. 459–468 (2000)Google Scholar
  13. 13.
    Somei, K., Kaneda, S., Fujii, T., Murata, S.: A microfluidic device for DNA tile self-assembly. In: DNA Computing (DNA 11), pp. 325–335 (2006)Google Scholar
  14. 14.
    Winfree, E.: Algorithmic self-assembly of DNA. Ph.D thesis, California Institute of Technology (1998)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Sándor P. Fekete
    • 2
  • Christian Scheffer
    • 2
  • Arne Schmidt
    • 2
  1. 1.CSAILMITCambridgeUSA
  2. 2.Department of Computer ScienceTU BraunschweigBraunschweigGermany

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