Optimal Program-Size Complexity for Self-Assembled Squares at Temperature 1 in 3D

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 program-size complexity of $O\left( \frac{\log N}{\log \log N}\right)$ (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). Note that this tile complexity is optimal for all algorithmically random values of N. 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 et al. (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 (SIAM J Comput 36(6):1544–1569, 2007), and perhaps of independent interest. We also develop a more sophisticated 3D temperature 1 optimal encoding construction that places asymptotically fewer tiles than our original optimal encoding construction. We use the method of conditional determinism of Lutz and Shutters (Theory Comput Syst 51(3):372–400, 2012) to prove the correctness of our constructions.