We augment a tree T with a shortcut pq to minimize the largest distance between any two points along the resulting augmented tree $T+pq$. We study this problem in a continuous and geometric setting where T is a geometric tree in the Euclidean plane, a shortcut is a line segment connecting any two points along the edges of T, and we consider all points on $T+pq$ (i.e., vertices and points along edges) when determining the largest distance along $T+pq$. The continuous diameter is the largest distance between any two points along edges. We establish that a single shortcut is sufficient to reduce the continuous diameter of a geometric tree T if and only if the intersection of all diametral paths of T is neither a line segment nor a point. We determine an optimal shortcut for a geometric tree with n straight-line edges in $O(n\log n)$ time.

我们用快捷方式pq扩展树T，以最小化沿结果扩展树的任意两点之间的最大距离$Ť+pq$。我们在连续的几何设置中研究此问题，其中T是欧几里得平面中的几何树，捷径是沿着T的边缘连接任意两个点的线段，我们考虑了所有点$Ť+pq$ （即沿边的顶点和点）确定沿时的最大距离 $Ť+pq$。的连续直径是沿着边缘的任何两点之间的最大距离。我们建立了一个捷径足以减小几何树T的连续直径，且前提是T的所有直径路径的交点既不是线段也不是点。我们确定具有n个直线边的几何树的最佳捷径$Ø（ñ\mathrm{日志}ñ）$ 时间。