Given a plane undirected graph $G$ with non-negative edge weights and a set of $k$ terminal pairs on the external face, it is shown in Takahashi et al., (Algorithmica, 16, 1996, pp. 339-357) that the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs (if they exist) can be computed in $O(n \log n)$ worst-case time, where $n$ is the number of vertices of $G$. This technique only applies when the union $U$ of the computed shortest paths is a forest. We show that given a plane undirected weighted graph $U$ and a set of $k$ terminal pairs on the external face, it is always possible to compute the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs in linear worst-case time, provided that the graph $U$ is the union of $k$ shortest paths, possibly containing cycles. Moreover, each shortest path $\pi$ can be listed in $O(\ell+\ell\log\lceil{\frac{k}{\ell}}\rceil)$, where $\ell$ is the number of edges in $\pi$. As a consequence, the problem of computing multi-terminal distances in a plane undirected weighted graph can always be solved in $O(n \log k)$ worst-case time in the general case.

翻译：平面上没有方向的平面图$G$(如果存在的话),其长度可以用美元(n\log n)来计算,最坏的时段是美元是美元是美元。这一技术只适用于计算最短路径的联盟美元是一个森林。我们显示,如果平面上没有方向的加权美元,美元是美元,外部是美元,一套美元是339-357美元,那么,如果存在美元,那么在最坏的时段上,非跨的最短路径的美元长度可以以美元计算,而美元是美元。如果美元是最短路径的联盟美元,那么计算最短路径的联盟美元是美元,计算最短路径的联盟美元是森林。我们显示,如果平面上没有方向的加权美元,美元是美元,那么在外部的平面上一套美元,则总是有可能用美元来计算,在最坏的时段里可以计算到美元与美元终端对美元,只要美元是最短路径的联盟美元,可能包含周期。此外,每条最短路径$\美元,一般的平面的平面是美元。