Non-crossing Shortest Paths Lengths in Planar Graphs in Linear Time

Abstract

Given a plane graph it is known how to compute the union of non-crossing shortest paths. These algorithms do not allow neither to list each single shortest path nor to compute length of shortest paths. Given the union of non-crossing shortest paths, we introduce the concept of shortcuts that allows us to establish whether a path is a shortest path by checking local properties on faces of the graph. By using shortcuts we can compute the length of each shortest path, given their union, in total linear time, and we can list each shortest path p in $$O(\max \{\ell , \ell \log \log \frac{k}{\ell } \})$$ , where $$\ell $$ is the number of edges in p and k the number of shortest paths.