Abstract
In this paper we design tile self-assembly systems which assemble arbitrarily close approximations to target squares with arbitrarily high probability. This is in contrast to previous work which has only considered deterministic assemblies of a single shape. Our technique takes advantage of the ability to assign tile concentrations to each tile type of a self-assembly system. Such an assignment yields a probability distribution over the set of possible assembled shapes. We show that by considering the assembly of close approximations to target shapes with high probability, as opposed to exact deterministic assembly, we are able to achieve significant reductions in tile complexity. In fact, we restrict ourselves to constant sized tile systems, encoding all information about the target shape into the tile concentration assignment. In practice, this offers a potentially useful tradeoff, as large libraries of particles may be infeasible or require substantial effort to create, while the replication of existing particles to adjust relative concentration may be much easier. To illustrate our technique we focus on the assembly of n×n squares, a special case class of shapes whose study has proven fruitful in the development of new self-assembly systems.
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References
Adleman, L., Cheng, Q., Goel, A., Huang, M.: Running time and program size for self-assembled squares. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 740–748 (2001)
Adleman, L., Cheng, Q., Goel, A., Huang, M., Kempe, D., Espanes, P., Rothemund, P.: Combinatorial optimization problems in self-assembly. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 23–32 (2002)
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 Journal on Computing 34, 1493–1515 (2005)
Aggarwal, G., Goldwasser, M.H., Kao, M.-Y., Schweller, R.T.: Complexities for generalized models of self-assembly. In: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 880–889 (2004)
Becker, F., Remila, E., Rapaport, I.: Self-assemblying classes of shapes with a minimum number of tiles, and in optimal time. In: Proceedings of the 26th Conference on Foundations of Software Technology and Theoretical Computer Science (2006)
Demaine, E., Demaine, M., Fekete, S., Ishaque, M., Rafalin, E., Schweller, R., Souvaine, D.: Staged self-assembly: Nanomanufacture of arbitrary shapes with O(1) glues. In: Proceedings of the 13th International Meeting on DNA Computing (2007)
Fu, T.-J., Seeman, N.C.: DNA double-crossover molecules. Biochemistry 32, 3211–3220 (1993)
Kao, M.-Y., Schweller, R.: Reducing tile complexity for self-assembly through temperature programming. In: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithms, pp. 571–580 (2006)
LaBean, T.H., Yan, H., Kopatsch, J., Liu, F., Winfree, E., Reif, H.J., Seeman, N.C.: The construction, analysis, ligation and self-assembly of DNA triple crossover complexes. J. Am. Chem. Soc. 122, 1848–1860 (2000)
Lagoudakis, M.G., Labean, T.H.: 2D DNA self-assembly for satisfiability. In: Proceedings of the 5th DIMACS Workshop on DNA Based Computers, June 26 1999, pp. 459–468 (1999)
Reif, J.: Local parallel biomolecular computation. In: Proceedings of the 3rd Annual DIMACS Workshop on DNA Based Computers, June 23-26 (1997)
Rothemund, P., Winfree, E.: The program-size complexity of self-assembled squares. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 459–468 (2000)
Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. In: Tenth International Meeting on DNA Computing, pp. 344–354 (2005)
Wang, H.: Proving theorems by pattern recognition. Bell System Technical Journal 40, 1–42 (1961)
Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, Pasadena (1998)
Winfree, E., Liu, F., Wenzler, L., Seeman, N.: Design and self-assembly of two-dimensional DNA crystals. Nature 394, 539–544 (1998)
Winfree, E., Yang, X., Seeman, N.C.: Universal computation via self-assembly of DNA: Some theory and experiments. In: Proceedings of the 2nd International Meeting on DNA Based Computers, June 10-12 1996, pp. 191–213 (1996)
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Kao, MY., Schweller, R. (2008). Randomized Self-assembly for Approximate Shapes. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_31
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DOI: https://doi.org/10.1007/978-3-540-70575-8_31
Publisher Name: Springer, Berlin, Heidelberg
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