We investigate a fundamental question regarding a benchmark class of shapes in one of the simplest, yet most widely utilized abstract models of algorithmic tile self-assembly. More specifically, we study the directed tile complexity of a \(\varvec{k}\times \varvec{N}\) thin rectangle in Winfree’s ubiquitous abstract Tile Assembly Model, assuming that cooperative binding cannot be enforced (temperature-1 self-assembly) and that tiles are allowed to be placed at most one step into the third dimension (just-barely 3D). While the directed tile complexities of a square and a scaled-up version of any algorithmically specified shape at temperature 1 in just-barely 3D are both asymptotically the same as they are (respectively) at temperature 2 in 2D, the (nearly tight) bounds on the directed tile complexity of a thin rectangle at temperature 2 in 2D are not currently known to hold at temperature 1 in just-barely 3D. Motivated by this discrepancy, we establish new lower and upper bounds on the directed tile complexity of a thin rectangle at temperature 1 in just-barely 3D. The proof of our upper bound is based on the construction of a novel, just-barely 3D temperature-1 self-assembling counter. Each value of the counter is comprised of \(\varvec{k} - {\textbf {2}}\) digits, represented in a geometrically staggered fashion within \(\varvec{k}\) rows. This nearly optimal digit density, along with the base of the counter, which is proportional to \(\varvec{N}^{\frac{\varvec{1}}{\varvec{k}-{\textbf {1}}}}\), results in an upper bound on the directed tile complexity of a thin rectangle at temperature 1 in just-barely 3D of \(\varvec{O}\left( \varvec{N}^{\frac{{\textbf {1}}}{\varvec{k}-{\textbf {1}}}} + \varvec{k} \right) \), and is an asymptotic improvement over the previous state-of-the-art upper bound. On our way to proving our lower bound, we develop a new, more powerful type of specialized Window Movie Lemma that lets us bound the number of “sufficiently similar” ways to assign glues to a set (rather than a sequence) of fixed locations. Consequently, our lower bound on the directed tile complexity of a thin rectangle at temperature 1 in just-barely 3D of \(\varvec{\Omega }\left( \varvec{N}^{\frac{{\textbf {1}}}{\varvec{k}}} \right) \), is also an asymptotic improvement over the previous state-of-the-art lower bound.

我们研究了一个关于形状基准类的基本问题，这是最简单但使用最广泛的算法瓦片自组装抽象模型之一。更具体地说，我们研究了\(\varvec{k}\times \varvec{N}\)的有向瓦片复杂度Winfree 无处不在的抽象 Tile 组装模型中的细矩形，假设无法强制执行协作绑定（温度 1 自组装），并且允许将 Tile 最多放置到三维（仅 3D）中一步。虽然在 3D 温度 1 下，正方形和任何算法指定形状的放大版本的有向图块复杂度都渐近地与 2D 温度 2 下相同，但（几乎紧的）边界目前还不知道在 2D 温度下薄矩形的有向平铺复杂性是否在 3D 温度下保持在 1 温度下。受这种差异的启发，我们在 3D 温度为 1 的情况下，为薄矩形的有向瓦复杂度建立了新的下限和上限。我们的上限证明基于一种新颖的、仅 3D 温度 1 自组装计数器的构造。计数器的每个值由以下部分组成\(\varvec{k} - {\textbf {2}}\)位数字，在\(\varvec{k}\)行中以几何交错方式表示。这种接近最佳的数字密度以及计数器的基数与\(\varvec{N}^{\frac{\varvec{1}}{\varvec{k}-{\textbf {1}}成正比}}\)，导致温度为 1 的薄矩形的有向瓦片复杂度在\(\varvec{O}\left( \varvec{N}^{\frac{{\ textbf {1}}}{\varvec{k}-{\textbf {1}}}} + \varvec{k} \right) \)，并且是对先前最先进上限的渐近改进。在证明下限的过程中，我们开发了一种新的、更强大的专用窗口电影引理，它使我们能够限制“足够相似”的方法的数量，以将粘合分配给一组（而不是一系列）固定位置。因此，我们在\(\varvec{\Omega }\left( \varvec{N}^{\frac{{\textbf {1} }}{\varvec{k}}} \right) \)也是对之前最先进的下界的渐近改进。