Computing Lengths of Non-Crossing Shortest Paths in Planar Graphs

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 union $U$ of $k$ non-crossing shortest paths joining the $k$ terminal pairs (if they exist) can be computed in $O(n\log n)$ time, where $n$ is the number of vertices of $G$. In the restricted case in which the union $U$ of the shortest paths is a forest, it is also shown that their lengths can be computed in the same time bound. We show in this paper that it is always possible to compute the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs in linear time, once the shortest paths union $U$ has been computed, also in the case $U$ contains cycles. Moreover, each shortest path $\pi$ can be listed in $O(\max\{\ell, \ell \log\frac{k}{\ell} \})$, where $\ell$ is the number of edges in $\pi$. As a consequence, the problem of computing non-crossing shortest paths and their lengths in a plane undirected weighted graph can be solved in $O(n\log k)$ time in the general case.