随机自组装实现精确自组装
Randomized Self-Assembly for Exact Shapes
摘要：
工作在Winfree的抽象瓷砖组装模型，我们表明，一个恒定大小的瓷砖组装系统可以通过编程相对瓷砖浓度，以建立一个n × n平方与高概率，任何足够大的n。这回答了Kao和Schweller（Randomized Self-Assembly for Approximate Decomposition，ICALP 2008）的一个开放性问题，他们展示了如何使用瓷砖浓度编程构建一个近似n x n的正方形，并询问近似是否可以以高概率精确。我们展示了如何修改这种技术来回答Kao和Schweller的另一个问题，通过展示一个恒定大小的瓷砖组装系统可以通过瓷砖浓度来编程，以组装任意有限 * 缩放形状 *，这些形状是通过将每个点替换为c x c块点来修改的，对于某个整数c。此外，我们表现出一个平稳的权衡之间的指定位n通过瓦片浓度与指定它们通过硬编码的瓦片类型，这允许瓦片浓度编程被用于指定一小部分的位的“输入”到一个瓦片组装系统，在浓度的约束下，只能指定到一个有限的精度。最后，考虑到瓷砖浓度规划模型的一些不切实际的方面，我们展示了如何修改施工，只使用浓度是任意接近均匀。
英文摘要：
Working in Winfree's abstract tile assembly model, we show that a constant-size tile assembly system can be programmed through relative tile concentrations to build an n x n square with high probability, for any sufficiently large n. This answers an open question of Kao and Schweller (Randomized Self-Assembly for Approximate Shapes, ICALP 2008), who showed how to build an approximately n x n square using tile concentration programming, and asked whether the approximation could be made exact with high probability. We show how this technique can be modified to answer another question of Kao and Schweller, by showing that a constant-size tile assembly system can be programmed through tile concentrations to assemble arbitrary finite *scaled shapes*, which are shapes modified by replacing each point with a c x c block of points, for some integer c. Furthermore, we exhibit a smooth tradeoff between specifying bits of n via tile concentrations versus specifying them via hard-coded tile types, which allows tile concentration programming to be employed for specifying a fraction of the bits of "input" to a tile assembly system, under the constraint that concentrations can only be specified to a limited precision. Finally, to account for some unrealistic aspects of the tile concentration programming model, we show how to modify the construction to use only concentrations that are arbitrarily close to uniform.
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