转子-路由器图搜索的时空最优性
Time and space optimality of rotor-router graph exploration
摘要：
我们考虑一个匿名的，端口标记的，无向图的探索问题与$n$节点和$m$边和直径$D$，由一个单一的移动代理。最初，代理不知道图形拓扑结构，也不知道任何全局参数。此外，当进入顶点时，代理不知道传入端口。每个顶点都被赋予了内存，可以由代理在访问该节点时读取和修改。然而，代理没有操作存储器，即，它在遍历边时不能携带任何状态。在这样的模型中，在度$d$的每个顶点处至少需要$\log_2 d$位，以使代理能够遍历每个图边缘。这个比特数总是足以使用算法Rotor-Router在时间$O(mD)$上探索任何图。我们表明，即使可用的节点内存是无限的，然后时间$Ω(n^3)$有时需要任何算法。这表明Rotor-Router在最坏情况图中是渐近最优的。其次，我们表明，对于路径的情况下，转子路由器达到最佳的时间。
英文摘要：
We consider the problem of exploration of an anonymous, port-labeled, undirected graph with $n$ nodes and $m$ edges and diameter $D$, by a single mobile agent. Initially the agent does not know the graph topology nor any of the global parameters. Moreover, the agent does not know the incoming port when entering to a vertex. Each vertex is endowed with memory that can be read and modified by the agent upon its visit to that node. However the agent has no operational memory i.e., it cannot carry any state while traversing an edge. In such a model at least $\log_2 d$ bits are needed at each vertex of degree $d$ for the agent to be able to traverse each graph edge. This number of bits is always sufficient to explore any graph in time $O(mD)$ using algorithm Rotor-Router. We show that even if the available node memory is unlimited then time $Ω(n^3)$ is sometimes required for any algorithm. This shows that Rotor-Router is asymptotically optimal in the worst-case graphs. Secondly we show that for the case of the path the Rotor-Router attains exactly optimal time.
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。
