Efficient tile sets for self assembling rectilinear shapes is of critical importance in algorithmic self assembly. A lower bound on the tile complexity of any deterministic self assembly system for an n × n square is \(\Upomega(\frac{\log(n)}{\log(\log(n))})\) (inferred from the Kolmogrov complexity). Deterministic self assembly systems with an optimal tile complexity have been designed for squares and related shapes in the past. However designing \(\Uptheta(\frac{\log(n)}{\log(\log(n))})\) unique tiles specific to a shape is still an intensive task in the laboratory. On the other hand copies of a tile can be made rapidly using PCR (polymerase chain reaction) experiments. This led to the study of self assembly on tile concentration programming models. We present two major results in this paper on the concentration programming model. First we show how to self assemble rectangles with a fixed aspect ratio (α:β), with high probability, using \(\Uptheta(\alpha+\beta)\) tiles. This result is much stronger than the existing results by Kao et al. (Randomized self-assembly for approximate shapes, LNCS, vol 5125. Springer, Heidelberg, 2008) and Doty (Randomized self-assembly for exact shapes. In: proceedings of the 50th annual IEEE symposium on foundations of computer science (FOCS), IEEE, Atlanta. pp 85–94, 2009)—which can only self assembly squares and rely on tiles which perform binary arithmetic. On the other hand, our result is based on a technique called staircase sampling. This technique eliminates the need for sub-tiles which perform binary arithmetic, reduces the constant in the asymptotic bound, and eliminates the need for approximate frames (Kao et al. Randomized self-assembly for approximate shapes, LNCS, vol 5125. Springer, Heidelberg, 2008) . Our second result applies staircase sampling on the equimolar concentration programming model (The tile complexity of linear assemblies. In: proceedings of the 36th international colloquium automata, languages and programming: Part I on ICALP ’09, Springer-Verlag, pp 235–253, 2009), to self assemble rectangles (of fixed aspect ratio) with high probability. The tile complexity of our algorithm is \(\Uptheta(\log(n))\) and is optimal on the probabilistic tile assembly model (PTAM)—n being an upper bound on the dimensions of a rectangle.

中文翻译：

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在浓度规划和概率瓷砖组装模型上自组装矩形形状。

用于自组装直线形状的高效瓷砖集在算法自组装中至关重要。对于n  ×  n正方形，任何确定性自组装系统的瓦片复杂度的下限是\(\Upomega(\frac{\log(n)}{\log(\log(n))})\)（推断来自 Kolmogrov 复杂性）。过去，已经为正方形和相关形状设计了具有最佳瓷砖复杂性的确定性自组装系统。然而，设计\(\Uptheta(\frac{\log(n)}{\log(\log(n))})\)特定于形状的独特瓷砖仍然是实验室中的一项艰巨任务。另一方面，瓷砖的副本可以使用 PCR（聚合酶链反应）实验快速制作。这导致了对自组装的研究瓷砖浓度编程模型。我们在本文中介绍了关于浓度规划模型的两个主要结果。首先，我们展示了如何使用\(\Uptheta(\alpha+\beta)\)图块以高概率自组装具有固定纵横比 (α:β) 的矩形。这个结果比 Kao 等人现有的结果强得多。（近似形状的随机自组装，LNCS，第 5125 卷。施普林格，海德堡，2008 年）和 Doty（精确形状的随机自组装。在：第 50 届 IEEE 计算机科学基础 (FOCS) 基础研讨会论文集，IEEE , Atlanta. pp 85–94, 2009)——它只能自组装正方形并依赖于执行二进制算术的瓦片。另一方面，我们的结果基于称为阶梯采样的技术. 这种技术消除了对执行二进制算术的子瓦片的需要，减少了渐近界中的常数，并消除了对近似帧的需要（Kao 等人。随机自组装近似形状，LNCS，第 5125 卷。Springer，海德堡, 2008)。我们的第二个结果将阶梯采样应用于等摩尔浓度规划模型（线性组件的瓦片复杂性。在：第 36 届国际学术讨论会自动机、语言和编程的会议录：关于 ICALP '09 的第一部分，Springer-Verlag，第 235-253 页， 2009），以高概率自组装矩形（固定纵横比）。我们算法的瓦片复杂度为\(\Uptheta(\log(n))\)并且在概率瓦片组装模型 (PTAM) 上是最优的——n是矩形尺寸的上限。