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Randomized Self Assembly of Rectangular Nano Structures

机译：矩形纳米结构的随机自组装

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Self assembly systems have numerous critical applications in medicine, circuit design, etc. For example, they can serve as nano drug delivery systems. The problem of assembling squares has been well studied. A lower bound on the tile complexity of any deterministic self assembly system for an N × N square is Ω((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 Θ((log(N)/(log(log(N))) unique tiles specific to a shape which needs to be self assembled is still an intensive task. Creating a copy of a tile is much simpler than creating a unique tile. With this constraint in mind probabilistic self assembly systems were introduced. These systems have O(l) tile complexity and the concentration of each of the tiles can be varied to produce the desired shape. Becker, et al. [1] introduced a line sampling technique which can self assemble mxn rectangles, where m is the expected width and n is the expected height of the rectangle. Kao, et al. [2] combined the line sampling technique with binary counters in a novel way to self assemble a supertile which can encode a binary string. This supertile can then be used to produce an n' × n' square such that (1 - ∈)n < n' < (1 +∈)n (for some relevant e) with probability > 1 - δ for sufficiently large n (i.e., n > f(e,S), for some appropriate function f). Doty [3] made the idea of Kao more precise, however the underlying construction is still based on sub-tiles to perform binary counting and division. In this paper we present randomized algorithms that can self assemble squares, rectangles and rectangles with constant aspect ratio with high probability (i.e. Ω(l - la), for any fixed a > 0) where n is the dimension of the shape which needs to be self assembled. Our self assembly constructions do not need any approximation frames introduced in Kao et al. [2] and hence are much cleaner and has significantly smaller constant in the tile complexity compared to both Kao [2] and Doty [3]. Finally In contrast to the existing randomized self assembly techniques our techniques can also self assemble a much stronger class of rectangles which have a fixed aspect ratio (α/β).
机译：自组装系统在医学，电路设计等方面具有许多关键应用。例如，它们可以用作纳米药物输送系统。组装正方形的问题已得到充分研究。对于N×N正方形，任何确定性自组装系统的图块复杂度的下限是Ω（（log（N）/（log（log（N））））（从Kolmogrov复杂度推导）。过去曾针对正方形和相关形状设计了最佳的瓷砖复杂度，但是针对特定形状（需要自组装）设计Θ（（log（N）/（log（log（N））））唯一瓷砖仍然是一个难题。繁琐的任务。创建一个瓷砖的副本比创建一个唯一的瓷砖要简单得多。考虑到这一约束，引入了概率自组装系统。这些系统具有O（l）个瓷砖的复杂性，每个瓷砖的浓度可以改变Becker等人[1]引入了一种可以自组装mxn矩形的线采样技术，其中m是期望的宽度，n是期望的矩形高度Kao等人[2]以一种新颖的方式将行采样技术与二进制计数器相结合，以自组装电源可以编码二进制字符串的rtile。然后可以使用该上位块来生成n'×n'的平方，使得对于足够大的n（（1-∈）n 1-δ。即，对于某些适当的函数f），n> f（e，S）。 Doty [3]使Kao的概念更加精确，但是底层结构仍然基于子区块来执行二进制计数和除法。在本文中，我们提出了一种随机算法，该算法可以以高概率（即，对于任何固定a> 0的Ω（l / na），对于恒定的宽高比）自组装正方形，矩形和矩形，其中n是形状的尺寸，需要自组装。我们的自组装结构不需要Kao等人介绍的任何近似框架。 [2]因此比Kao [2]和Doty [3]都更清洁，并且瓷砖复杂度常数小得多。最终，与现有的随机自组装技术相反，我们的技术还可以自组装具有固定长宽比（α/β）的一类更强的矩形。

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