Complexities for Generalized Models of Self-assembly

by Gagan Aggarwal, Qi Cheng, Michael H. Goldwasser, Ming-Yang Kao, Pablo Moisset de Espanes, and Robert T. Schweller

Abstract:

In this paper, we study the complexity of self-assembly under models that are natural generalizations of the tile self-assembly model. In particular, we extend Rothemund and Winfree's study of the tile complexity of tile self-assembly [Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, Portland, OR, 2000, pp. 459--468]. They provided a lower bound of Ω(log N / log log N) on the tile complexity of assembling an N x N square for almost all N. Adleman~et~al. [Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Greece, 2001, pp. 740--748] gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size O(sqrt(log N)) which assembles an N x N square in a model which allows flexible glue strength between nonequal glues. This result is matched for almost all N by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the Ω(log N / log log N) lower bound applies to N x N squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length N and width k, we provide a tighter lower bound of Ω(N1/k/k) for the standard model, yet we also give a construction which achieves O(log N / log log N) complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape; we show that this problem is NP-hard for three of the generalized models.


Citation:
Complexities for Generalized Models of Self-assembly
Gagan Aggarwal, Qi Cheng, Michael H. Goldwasser, Ming-Yang Kao, Pablo Moisset de Espanes, and Robert T. Schweller
SIAM Journal of Computing, 34(6):1493-1515, 2005.
DOI:10.1137/S0097539704445202
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A preliminary version originally appeared as:
Complexities for Generalized Models of Self-Assembly
Gagan Aggarwal, Michael H. Goldwasser, Ming-Yang Kao, and Robert T. Schweller
Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), New Orleans, Louisiana, Jan. 2004, pp. 880-889.
DOI:10.1145/982792.982926
Download as: [ abstract, pdf, pdf.gz, pdf.Z ]

Michael Goldwasser