An introduction to Kolmogorov complexity and its applications (2nd ed.)
An introduction to Kolmogorov complexity and its applications (2nd ed.)
The program-size complexity of self-assembled squares (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Running time and program size for self-assembled squares
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Complexities for generalized models of self-assembly
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Complexity of Self-Assembled Shapes
SIAM Journal on Computing
Randomized Self-assembly for Approximate Shapes
ICALP '08 Proceedings of the 35th international colloquium on Automata, Languages and Programming, Part I
Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues
DNA13'07 Proceedings of the 13th international conference on DNA computing
Self-assemblying classes of shapes with a minimum number of tiles, and in optimal time
FSTTCS'06 Proceedings of the 26th international conference on Foundations of Software Technology and Theoretical Computer Science
Error free self-assembly using error prone tiles
DNA'04 Proceedings of the 10th international conference on DNA computing
Randomized Self-Assembly for Exact Shapes
SIAM Journal on Computing
Parallelism and time in hierarchical self-assembly
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The power of nondeterminism in self-assembly
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Natural Computing: an international journal
Self-assembly with geometric tiles
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Theory of algorithmic self-assembly
Communications of the ACM
One-dimensional staged self-assembly
Natural Computing: an international journal
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The conventional Tile Assembly Model (TAM) developed by Winfree using Wang tiles is a powerful, Turing-universal theoretical framework which models varied self-assembly processes. We describe a natural extension to TAM called the Probabilistic Tile Assembly Model (PTAM) to model the inherent probabilistic behavior in physically realized self-assembled systems. A particular challenge in DNA nanoscience is to form linear assemblies or rulers of a specified length using the smallest possible tile set. These rulers can then be used as components for construction of other complex structures. In TAM, a deterministic linear assembly of length N requires a tile set of cardinality at least N . In contrast, for any given N , we demonstrate linear assemblies of expected length N with a tile set of cardinality *** (logN ) and prove a matching lower bound of *** (logN ). We also propose a simple extension to PTAM called *** -pad systems in which we associate *** pads with each side of a tile, allowing abutting tiles to bind when at least one pair of corresponding pads match and prove analogous results. All our probabilistic constructions are free from co-operative tile binding errors and can be modified to produce assemblies whose probability distribution of lengths has arbitrarily small tail bounds dropping exponentially with a given multiplicative factor increase in number of tile types. Thus, for linear assembly systems, we have shown that randomization can be exploited to get large improvements in tile complexity at a small expense of precision in length.