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.