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
Computation by Self-assembly of DNA Graphs
Genetic Programming and Evolvable Machines
Algorithmic self-assembly of dna
Algorithmic self-assembly of dna
Complexities for generalized models of self-assembly
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Complexities for Generalized Models of Self-Assembly
SIAM Journal on Computing
Complexity of Self-Assembled Shapes
SIAM Journal on Computing
Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues
Natural Computing: an international journal
The Tile Complexity of Linear Assemblies
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Polyomino-Safe DNA Self-assembly via Block Replacement
DNA Computing
Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
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
Compact error-resilient computational DNA tiling assemblies
DNA'04 Proceedings of the 10th international conference on DNA computing
An introduction to tile-based self-assembly
UCNC'12 Proceedings of the 11th international conference on Unconventional Computation and Natural Computation
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In this work we propose a generalization of Winfree's abstract Tile Assembly Model (aTAM) in which tile types are assigned rigid shapes, or geometries, along each tile face. We examine the number of distinct tile types needed to assemble shapes within this model, the temperature required for efficient assembly, and the problem of designing compact geometric faces to meet given compatibility specifications. We pose the following question: can complex geometric tile faces arbitrarily reduce the number of distinct tile types to assemble shapes? Within the most basic generalization of the aTAM, we show that the answer is no. For almost all n at least $\Omega(\sqrt{\log n})$ tile types are required to uniquely assemble an n×n square, regardless of how much complexity is pumped into the face of each tile type. However, we show for all n we can achieve a matching $O(\sqrt{\log n})$ tile types, beating the known lower bound of Θ(logn / loglogn) that holds for almost all n within the aTAM. Further, our result holds at temperature τ=1. Our next result considers a geometric tile model that is a generalization of the 2-handed abstract tile assembly model in which tile aggregates must move together through obstacle free paths within the plane. Within this model we present a novel construction that harnesses the collision free path requirement to allow for the unique assembly of any n×n square with a sleek O(loglogn) distinct tile types. This construction is of interest in that it is the first tile self-assembly result to harness collision free planar translation to increase efficiency, whereas previous work has simply used the planarity restriction as a desireable quality that could be achieved at reduced efficiency. This surprisingly low tile type result further emphasizes a fundamental open question: Is it possible to assemble n×n squares with O(1) distinct tile types? Essentially, how far can the trade off between the number of distinct tile types required for an assembly and the complexity of each tile type itself be taken?