可重构电路下变形虫模型中的形状包含问题
On the Shape Containment Problem within the Amoebot Model with Reconfigurable Circuits
AI总结：
本文在几何变形虫模型中利用可重构电路扩展研究形状包含问题，证明了$Ω(\sqrt{n})$的通用下界，并为雪花形状和星凸形状分别提出了$\mathcal{O}(\sqrt{n} \log n)$和$\mathcal{O}(\log^2 k)$轮次的求解算法。
摘要：
在可编程物质中，我们考虑大量被称为粒子的微小原始计算实体，它们运行分布式算法来控制粒子结构的全局属性。形状形成问题，即粒子必须使用基本移动能力将自身重组为所需形状，尤为有趣。在相关的形状包含问题中，粒子被给定一个形状$S$的描述，必须在初始配置中找到$S$的最大缩放表示，且不进行移动。尽管形状形成问题已被广泛研究，但形状包含问题却未受关注，它除了形状形成外可能还有检测结构缺陷等额外用途。本文在用于可编程物质的几何变形虫模型中考虑形状包含问题，利用其可重构电路扩展来实现粒子连通子集上原始信号的瞬时传输。我们首先证明了一般问题在同步轮次上的下界为$Ω(\sqrt{n})$，其中$n$是粒子数量。然后，我们构造了雪花形状类及其子类星凸形状，并分别为两者提出了解决方案。设$k$为给定变形虫结构中所考虑形状的最大缩放比例。若该形状为星凸，我们在$\mathcal{O}(\log^2 k)$轮次内解决；若为雪花但非星凸，则在$\mathcal{O}(\sqrt{n} \log n)$轮次内解决。
英文摘要：
In programmable matter, we consider a large number of tiny, primitive computational entities called particles that run distributed algorithms to control global properties of the particle structure. Shape formation problems, where the particles have to reorganize themselves into a desired shape using basic movement abilities, are particularly interesting. In the related shape containment problem, the particles are given the description of a shape $S$ and have to find maximally scaled representations of $S$ within the initial configuration, without movements. While the shape formation problem is being studied extensively, no attention has been given to the shape containment problem, which may have additional uses beside shape formation, such as detection of structural flaws. In this paper, we consider the shape containment problem within the geometric amoebot model for programmable matter, using its reconfigurable circuit extension to enable the instantaneous transmission of primitive signals on connected subsets of particles. We first prove a lower runtime bound of $Ω(\sqrt{n})$ synchronous rounds for the general problem, where $n$ is the number of particles. Then, we construct the class of snowflake shapes and its subclass of star convex shapes, and present solutions for both. Let $k$ be the maximum scale of the considered shape in a given amoebot structure. If the shape is star convex, we solve it within $\mathcal{O}(\log^2 k)$ rounds. If it is a snowflake but not star convex, we solve it within $\mathcal{O}(\sqrt{n} \log n)$ rounds.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
