通过旋转实现的中心化保持连通性变换：所有正交凸矩阵的3个火枪手
Centralised Connectivity-Preserving Transformations by Rotation: 3 Musketeers for all Orthogonal Convex Shapes
摘要：
我们研究了一个可编程物质系统的模型，该系统由位于二维正方形网格上的$n$设备组成，这些设备能够执行相互旋转的最小机械操作。目标是将初始形状A转换为目标形状B。我们感兴趣的是，在这种情况下，在保持全球连通性的额外约束下，描述可以相互转化的形状。这是$[$ Michail等人留下的主要问题之一，JCSS'19 $]$。请注意，所考虑的问题是关于变换的结构可行性，我们专门通过集中式构造性证明来处理。分布式解决方案留给未来的工作，并形成一个有趣的研究方向。过去的工作取得了一些进展，为特殊类的漂亮的形状。我们在这里考虑一类正交凸形，其中对于网格上水平或垂直线上的任何两个节点$u, v$，在$u$和$v$之间没有空单元。我们开发了一个通用的集中式变换，并证明了，对于任何一对$A$，$B$颜色一致的正交凸形，它可以转换$A$到$B$。鉴于所考虑的类中存在块形状，我们使用最小的3节点种子来触发转换。我们的转换的运行时间是一个最佳的$O(n^2)$顺序移动，其中$n=|A|=|B|$。我们离开作为一个开放的问题，存在一个普遍的连接性保持转换的一个小种子。我们的信念是，本文中开发的技术可能会被证明有助于回答这个问题。
英文摘要：
We study a model of programmable matter systems consisting of $n$ devices lying on a 2-dimensional square grid, which are able to perform the minimal mechanical operation of rotating around each other. The goal is to transform an initial shape A into a target shape B. We are interested in characterising the class of shapes which can be transformed into each other in such a scenario, under the additional constraint of maintaining global connectivity at all times. This was one of the main problems left open by $[$Michail et al., JCSS'19$]$. Note that the considered question is about structural feasibility of transformations, which we exclusively deal with via centralised constructive proofs. Distributed solutions are left for future work and form an interesting research direction. Past work made some progress for the special class of nice shapes. We here consider the class of orthogonal convex shapes, where for any two nodes $u, v$ in a horizontal or vertical line on the grid, there is no empty cell between $u$ and $v$. We develop a generic centralised transformation and prove that, for any pair $A$, $B$ of colour-consistent orthogonal convex shapes, it can transform $A$ into $B$. In light of the existence of blocked shapes in the considered class, we use a minimal 3-node seed to trigger the transformation. The running time of our transformation is an optimal $O(n^2)$ sequential moves, where $n=|A|=|B|$. We leave as an open problem the existence of a universal connectivity-preserving transformation with a small seed. Our belief is that the techniques developed in this paper might prove useful to answer this.
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。
