For an ordered point set in a Euclidean space or, more generally, in an abstract metric space, the ordered Nearest Neighbor 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 Nearest Neighbor 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 Nearest Neighbor 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，有序最近邻图通过将有向边从每个点连接到其最近的前驱点而得到。我们证明，对于 $\mathbb{R}^d$ 中的任意 $n$ 个点集，总存在一种排序使得对应的有序最近邻图的最大度至少为 $\log{n}/(4d)$。除了 $1/(4d)$ 因子外，该界限是最优的。在抽象设置中，我们证明对于任意 $n$ 元度量空间，总存在一种排序使得对应的有序最近邻图具有 $\Omega(\sqrt{\log{n}/\log\log{n}})$ 的最大度。