Abstract
Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly α is producible. The algorithm can be optimized to use O(|α| log2 |α|) time. Cannon, Demaine, Demaine, Eisenstat, Patitz, Schweller, Summers, and Winslow [4] showed that the problem of deciding if an assembly α is the unique producible terminal assembly of a tile system \(\mathcal{T}\) can be solved in \(O(|\alpha|^2 |\mathcal{T}| + |\alpha| |\mathcal{T}|^2)\) time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to \(O(|\alpha| |\mathcal{T}| \log |\mathcal{T}|)\) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently – i.e., if the positions that they share have the same tile type in each assembly – then their union is also producible.
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References
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, STOC 2002, pp. 23–32 (2002)
Aggarwal, G., Cheng, Q., Goldwasser, M.H., Kao, M.-Y., de Espanés, P.M., Schweller, R.T.: Complexities for generalized models of self-assembly. SIAM Journal on Computing 34, 1493–1515 (2004); Preliminary version appeared in SODA 2004
Barish, R.D., Schulman, R., Rothemund, P.W.K., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proceedings of the National Academy of Sciences 106(15), 6054–6059 (2009)
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: Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science, STACS 2013, pp. 172–184 (2013)
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 (2012)
Thomas, H., Cormen, C.E., Leiserson, R.L.: Rivest, and Clifford Stein. In: Introduction to Algorithms. MIT Press (2001)
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. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 400–412. Springer, Heidelberg (2013)
Doty, D.: Theory of algorithmic self-assembly. Communications of the ACM 55(12), 78–88 (2012)
Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R.T., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, pp. 302–310. IEEE (2012)
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. IEEE (2010)
Hopcroft, J., Tarjan, R.: Algorithm 447: Efficient algorithms for graph manipulation. Communications of the ACM 16(6), 372–378 (1973)
Luhrs, C.: Polyomino-safe DNA self-assembly via block replacement. Natural Computing 9(1), 97–109 (2008), Preliminary version appeared in DNA 2008
Patitz, M.J.: An introduction to tile-based self-assembly. In: Durand-Lose, J., Jonoska, N. (eds.) UCNC 2012. LNCS, vol. 7445, pp. 34–62. Springer, Heidelberg (2012)
Schulman, R., Winfree, E.: Synthesis of crystals with a programmable kinetic barrier to nucleation. Proceedings of the National Academy of Sciences 104(39), 15236–15241 (2007)
Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. SIAM Journal on Computing 39(4), 1581–1616 (2009), Preliminary version appeared in DNA 2004
Winfree, E.: Simulations of computing by self-assembly. Technical Report CaltechCSTR:1998.22. Institute of Technology, California (1998)
Winfree, E.: Self-healing tile sets. In: Chen, J., Jonoska, N., Rozenberg, G. (eds.) Nanotechnology: Science and Computation. Natural Computing Series, pp. 55–78. Springer (2006)
Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394(6693), 539–544 (1998)
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Doty, D. (2014). Producibility in Hierarchical Self-assembly. In: Ibarra, O., Kari, L., Kopecki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2014. Lecture Notes in Computer Science(), vol 8553. Springer, Cham. https://doi.org/10.1007/978-3-319-08123-6_12
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DOI: https://doi.org/10.1007/978-3-319-08123-6_12
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