Let T be a tree with n vertices in a metric space. We consider the problem of adding one shortcut edge to T to minimize the radius of the resulting graph.

For the continuous version of the problem where a center may be a point in the interior of an edge of the graph we give a linear time algorithm. In the case when the center is restricted to lie on a vertex, the discrete version, we give an $O(n\log n)$ expected time algorithm.

Previously linear-time algorithms were known for the special case when the input graph is a path.

中文翻译：

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度量空间中半径最优增广树的算法

设T是度量空间中具有n 个顶点的树。我们考虑向T添加一条快捷边以最小化结果图的半径的问题。

对于中心可能是图形边缘内部的点的问题的连续版本，我们给出了线性时间算法。在中心被限制在一个顶点上的情况下，离散版本，我们给出一个$欧(n\mathrm{日志}n)$期望时间算法。

以前的线性时间算法因输入图是路径的特殊情况而为人所知。