温度为1的3D自组装的最佳程序大小复杂性
Optimal program-size complexity for self-assembly at temperature 1 in 3D
摘要：
工作在一个三维的变体Winfree的抽象瓷砖组装模型，我们表明，对于所有的$N \in \mathbb{N}$，有一个瓷砖集，唯一的自组装成一个$N \times N$正方形在温度1与最佳程序大小的复杂性$O(\log N / \log \log N)$（程序大小的复杂性，也称为瓷砖复杂性，形状是唯一的瓷砖类型所需的最小数量的唯一自组装它）。此外，我们的构造是“刚刚好”的3D，在这个意义上，它的作品，即使当瓷砖的位置被限制在$z = 0$和$z = 1$平面。这一结果肯定地回答了Cook，Fu，Schweller提出的一个未决问题（SODA 2011）。为了实现这一结果，我们开发了一个通用的3D温度1最佳编码结构，让人想起Soloveichik和Winfree（SICOMP 2007）的2D温度2最佳编码结构，也许是独立的兴趣。
英文摘要：
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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