Let P be a path graph of n vertices embedded in a metric space. We consider the problem of adding a new edge to P to minimize the radius of the resulting graph. Previously, a similar problem for minimizing the diameter of the graph was solved in $O(n\log n)$ time. To the best of our knowledge, the problem of minimizing the radius has not been studied before. In this paper, we present an $O(n)$ time algorithm for the problem, which is optimal.

中文翻译：

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用于度量空间中半径优化扩充路径的线性时间算法

令P为度量空间中嵌入的n个顶点的路径图。我们考虑了向P添加新边以最小化结果图半径的问题。以前，解决了最小化图形直径的类似问题$Ø（ñ\mathrm{日志}ñ）$时间。据我们所知，最小化半径的问题以前从未研究过。在本文中，我们提出了一个$Ø（ñ）$ 问题的时间算法，这是最优的。