Computing the Maximum Detour of a Plane Geometric Graph in Subquadratic Time

Let G be a plane 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 p and q of G of the ratio between the distance between p and q in G and the Euclidean distance ‖pq‖. 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) expected running time. We also show that if G has bounded treewidth, its maximum detour can be computed in O(n log n) expected time.