论通用型转子铣刀的分类
On the Classification of Universal Rotor-Routers
摘要：
转子-路由器的组合理论与统计力学、图论、混沌理论和计算机科学的问题有联系。一个转子路由器网络定义了一个有向图G上的确定性行走，其中一个粒子从一个源顶点行走，直到它到达几个目标顶点之一。受Giacaglia等人最近研究结果的启发，我们研究了所有非目标顶点具有相同类型的转子路由器网络。r型转子是通用的，如果每个命中序列可以通过完全由r型转子组成的同构转子路由器网络来实现。我们给出了一个对通用转子类型进行完全分类的猜想。然后，这个问题被简化的定理，我们称之为减少定理，使我们能够考虑只有两个状态的转子。一个转子路由器网络称为压缩机，因为它往往会缩短转子周期，随着相关的算法，确定几乎所有的转子的通用性。新的转子类，包括boppy转子，平衡转子，和BURD转子，被定义为严格研究该算法。使用压缩机的通用性的新的转子类被证明，和经验的计算机结果来支持我们的结论。在这些结果之前，已知长度最多为17的大约260，000种可能的双状态转子类型中，只有小于100种是通用的，而压缩机算法证明了除了272种转子类型之外的所有转子类型的通用性。
英文摘要：
The combinatorial theory of rotor-routers has connections with problems of statistical mechanics, graph theory, chaos theory, and computer science. A rotor-router network defines a deterministic walk on a digraph G in which a particle walks from a source vertex until it reaches one of several target vertices. Motivated by recent results due to Giacaglia et al., we study rotor-router networks in which all non-target vertices have the same type. A rotor type r is universal if every hitting sequence can be achieved by a homogeneous rotor-router network consisting entirely of rotors of type r. We give a conjecture that completely classifies universal rotor types. Then, this problem is simplified by a theorem we call the Reduction Theorem that allows us to consider only two-state rotors. A rotor-router network called the compressor, because it tends to shorten rotor periods, is introduced along with an associated algorithm that determines the universality of almost all rotors. New rotor classes, including boppy rotors, balanced rotors, and BURD rotors, are defined to study this algorithm rigorously. Using the compressor the universality of new rotor classes is proved, and empirical computer results are presented to support our conclusions. Prior to these results, less than 100 of the roughly 260,000 possible two-state rotor types of length up to 17 were known to be universal, while the compressor algorithm proves the universality of all but 272 of these rotor types.
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