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，抽象度量空间中的一个有序点集，有序最近邻图是通过将每个点与其最近的前驱点用一条有向边连接而得到的。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 $1/(4d)$ 这个因子外，该界是最优的。至于抽象设置，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。