Computing the maximum detour of a plane geometric graph in subquadratic time

Let G be a plane geometric graph where each edge is a line segment. We consider the problem of computing the maximum detour of G, defined as the maximum over all pairs of distinct points (vertices as well as interior points of edges) p and q of G of the ratio between the distance between p and q in G and the Euclidean distance ||pq||2. The fastest known algorithm for this problem has Θ(n2) running time where n is the number of vertices. We show how to obtain O(n3/2(log n)3) expected running time. We also show that if G has bounded treewidth, its maximum detour can be computed in O(n(log n)3) expected time.