For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Yao graph is obtained by connecting each of the points to its closest predecessor by a directed edge. We show that for every set of $n$ points in $\mathbb{R}^d$, there exists an order such that the corresponding ordered Yao graph has maximum degree at least $\log{n}/(4d)$. Apart from the $1/(4d)$ factor, this bound is the best possible. As for the abstract setting, we show that for every $n$-element metric space, there exists an order such that the corresponding ordered Yao graph has maximum degree $\Omega(\sqrt{\log{n}/\log\log{n}})$.

翻译：对于欧几里得空间（或更一般地，抽象度量空间）中的有序点集，有序Yao图通过将每个点与其最近的前驱点用有向边连接而得到。我们证明，对于 $\mathbb{R}^d$ 中的任意 $n$ 个点集，存在一种顺序使得相应的有序Yao图的最大度至少为 $\log{n}/(4d)$。除了 $1/(4d)$ 因子外，该界是最优的。至于抽象情形，我们证明，对于任意 $n$ 元度量空间，存在一种顺序使得相应的有序Yao图的最大度为 $\Omega(\sqrt{\log{n}/\log\log{n}})$。