Algorithms for Distance Problems in Continuous Graphs

We study the problem of computing the diameter and the mean distance of a continuous graph, i.e., a connected graph where all points along the edges, instead of only the vertices, must be taken into account. It is known that for continuous graphs with $m$ edges these values can be computed in roughly $O(m^2)$ time. In this paper, we use geometric techniques to obtain subquadratic time algorithms to compute the diameter and the mean distance of a continuous graph for two well-established classes of sparse graphs. We show that the diameter and the mean distance of a continuous graph of treewidth at most $k$ can be computed in $O(n\log^{O(k)} n)$ time, where $n$ is the number of vertices in the graph. We also show that computing the diameter and mean distance of a continuous planar graph with $n$ vertices and $F$ faces takes $O(n F \log n)$ time.