We present a routing algorithm for the directed $\Theta_4$-graph, here denoted as the $\overrightarrow{\Theta_4}}$-graph, that computes a path between any two vertices $s$ and $t$ having length at most $17$ times the Euclidean distance between $s$ and $t$. To compute this path, at each step, the algorithm only uses knowledge of the location of the current vertex, its (at most four) outgoing edges, the destination vertex, and one additional bit of information in order to determine the next edge to follow. This provides the first known online, local, competitive routing algorithm with constant routing ratio for the $\Theta_4$-graph, as well as improving the best known upper bound on the spanning ratio of these graphs from $237$ to $17$. We also show that without this additional bit of information, the routing ratio increases to $\sqrt{290} \approx 17.03$.

翻译：我们为直接的 $Theta_ 4$- graph 提出了一个路径算法。 这里的表示是 $\ overrightrow_Theta_ 4 ⁇ _ $- graph, 计算任何两个顶点的路径, 美元与美元之间长度最多为17美元的欧几里特距离的一条路径。 要计算这条路径, 每一步, 算法只使用对当前顶点位置、 其( 最多四个) 向外边缘、 目的地和另外一点信息的了解, 以确定下一个边缘。 这为第一个已知的、 本地的、 竞争性的路径算法提供了固定路径比 $\ Theta_ 4$- graph 提供了一条路径比, 并改进了这些图宽度比上方的最佳已知上限, 从 237 美元 到 17 美元。 我们还显示, 没有这一额外的信息, 路径比则增加到 $\ sqrt{ 290}\ approx 17.03$。