Abstract
Working in a three-dimensional variant of Winfree’s abstract Tile Assembly Model, we show that, for all \(N \in \mathbb {N}\), there is a tile set that uniquely self-assembles into an \(N \times N\) square shape at temperature 1 with optimal program-size complexity of \(O(\log N / \log \log N)\) (the program-size complexity, also known as tile complexity, of a shape is the minimum number of unique tile types required to uniquely self-assemble it). Moreover, our construction is “just barely” 3D in the sense that it works even when the placement of tiles is restricted to the \(z = 0\) and \(z = 1\) planes. This result affirmatively answers an open question from Cook, Fu, Schweller (SODA 2011). To achieve this result, we develop a general 3D temperature 1 optimal encoding construction, reminiscent of the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree (SICOMP 2007), and perhaps of independent interest.
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Notes
- 1.
Technically, Rothemund and Winfree established the 2D self-assembly case, but their proof easily generalizes to 3D self-assembly.
- 2.
One subtle difference between our 3D definition of K and the original 2D definition of the tile complexity of an \(N \times N\) square, given by Rothemund and Winfree in [7], is that they assume a fully-connected final structure, whereas we do not.
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Furcy, D., Micka, S., Summers, S.M. (2015). Optimal Program-Size Complexity for Self-Assembly at Temperature 1 in 3D. In: Phillips, A., Yin, P. (eds) DNA Computing and Molecular Programming. DNA 2015. Lecture Notes in Computer Science(), vol 9211. Springer, Cham. https://doi.org/10.1007/978-3-319-21999-8_5
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