COMPUTING THE MAXIMUM DETOUR OF A PLANE GEOMETRIC GRAPH IN SUBQUADRATIC TIME

Abstract

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 ( n 2 ) running time where n is the number of vertices. We show how to obtain O ( n 3/2 (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.