平行旋转机器人行走的锁定问题
Lock-in Problem for Parallel Rotor-router Walks
摘要：
转子路由器模型，也称为Propp机，被引入作为随机游走的确定性替代方案。在这个模型中，一组相同的令牌最初被放置在图的节点处。每个节点保持传出弧的循环排序，并且在连续的回合期间，令牌沿着根据该排序以循环方式选择的弧传播。模型的行为是完全确定的。Yanovski等人（2003）证明了在任何具有m条边和直径$D$的图上的单个转子路由器行走稳定于在图的边集上的所有2m个有向弧的集合上的欧拉回路的遍历，并且系统的这种周期性行为在至多2mD步的初始瞬态阶段之后实现。实验研究了多个平行转子路由器的情况，导致Yanovski等人猜想，$k \textgreater{} 1$平行行走系统也在最多$2m$步的长度期间稳定。在这项工作中，我们反驳了这一猜想，表明并行转子路由器行走的周期实际上可以是超多项式的图的大小。在积极的一面，我们提供了一个表征并行路由器走的周期性行为，在结构属性的稳态称为子循环分解。这个属性为我们提供了有效检测给定系统配置是否对应于系统的瞬态或极限行为的工具。此外，我们提供了多项式上界的$O(m^4 D^2 + mD \log k)$和$O(m^5 k^2)$的步骤，它需要为系统稳定。因此，我们能够预测任何未来的行为系统使用的算法，需要多项式时间和空间。此外，我们发现，存在一个分离的单步行和多步行转子路由器系统的稳定时间，并且对于某些图形，后者可以是渐近较大的情况下，即使是$k = 2$散步。
英文摘要：
The rotor-router model, also called the Propp machine, was introduced as a deterministic alternative to the random walk. In this model, a group of identical tokens are initially placed at nodes of the graph. Each node maintains a cyclic ordering of the outgoing arcs, and during consecutive turns the tokens are propagated along arcs chosen according to this ordering in round-robin fashion. The behavior of the model is fully deterministic. Yanovski et al.(2003) proved that a single rotor-router walk on any graph with m edges and diameter $D$ stabilizes to a traversal of an Eulerian circuit on the set of all 2m directed arcs on the edge set of the graph, and that such periodic behaviour of the system is achieved after an initial transient phase of at most 2mD steps. The case of multiple parallel rotor-routers was studied experimentally, leading Yanovski et al. to the conjecture that a system of $k \textgreater{} 1$ parallel walks also stabilizes with a period of length at most $2m$ steps. In this work we disprove this conjecture, showing that the period of parallel rotor-router walks can in fact, be superpolynomial in the size of graph. On the positive side, we provide a characterization of the periodic behavior of parallel router walks, in terms of a structural property of stable states called a subcycle decomposition. This property provides us the tools to efficiently detect whether a given system configuration corresponds to the transient or to the limit behavior of the system. Moreover, we provide polynomial upper bounds of $O(m^4 D^2 + mD \log k)$ and $O(m^5 k^2)$ on the number of steps it takes for the system to stabilize. Thus, we are able to predict any future behavior of the system using an algorithm that takes polynomial time and space. In addition, we show that there exists a separation between the stabilization time of the single-walk and multiple-walk rotor-router systems, and that for some graphs the latter can be asymptotically larger even for the case of $k = 2$ walks.
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