改进了3D中非协作地唯一自组装薄矩形的瓦片复杂度的上下界
Improved lower and upper bounds on the tile complexity of uniquely self-assembling a thin rectangle non-cooperatively in 3D
摘要：
我们调查一个基本的问题，在一个最简单的基准类的形状，但最广泛使用的抽象模型的算法瓷砖自组装。具体来说，我们研究了有向瓷砖复杂性的$k \times N$薄矩形Winfree的抽象瓷砖组装模型，假设合作绑定不能强制执行（温度-1自组装），瓷砖被允许放置在最多一步到第三维（刚刚勉强3D）。虽然在温度为1的情况下，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在刚刚3D中都与它们在2D中的温度为2的情况下（分别）渐近相同，但在2D中的温度为2的情况下，薄矩形的有向图块复杂度的界限在刚刚3D中的温度为1时并不存在。出于这种差异，我们建立了新的下限和上限上的温度为1的薄矩形的有向瓦片的复杂性在刚刚三维。我们开发了一种新的，更强大的类型的窗口电影引理，让我们上限的数量“足够相似”的方式分配胶水到一组固定的位置。因此，我们的下限，$Ω\left(N^{\frac{1}{k}}\right)$，是一个渐进的改进，在以前的最佳下限，是更美观，因为它消除了$k$用于划分$N^{\frac{1}{k}}$。我们的上限的证明是基于一个刚刚3D的，温度为1的计数器，根据“数字区域”组织，与以前的最佳计数器相比，它为相同的目标矩形提供了大约50%的数字。这种数字密度的增加导致$O\left(N^{\frac{1}{\left\lfloor\frac{k}{2}\right\rfloor}}+\log N\right)$的上界，这是对先前最佳上界的渐进改进，并且大致是我们下界的平方。
英文摘要：
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. Specifically, we study the directed tile complexity of a $k \times N$ thin rectangle in Winfree's 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 bounds on the directed tile complexity of a thin rectangle at temperature 2 in 2D are not 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. We develop a new, more powerful type of Window Movie Lemma that lets us upper bound the number of "sufficiently similar" ways to assign glues to a set of fixed locations. Consequently, our lower bound, $Ω\left(N^{\frac{1}{k}}\right)$, is an asymptotic improvement over the previous best lower bound and is more aesthetically pleasing since it eliminates the $k$ that used to divide $N^{\frac{1}{k}}$. The proof of our upper bound is based on a just-barely 3D, temperature-1 counter, organized according to "digit regions", which affords it roughly fifty percent more digits for the same target rectangle compared to the previous best counter. This increase in digit density results in an upper bound of $O\left(N^{\frac{1}{\left\lfloor\frac{k}{2}\right\rfloor}}+\log N\right)$, that is an asymptotic improvement over the previous best upper bound and roughly the square of our lower bound.
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