Computing the dilation of edge-augmented graphs in metric spaces

Let G=(V,E) be an undirected graph with n vertices embedded in a metric space. We consider the problem of adding a shortcut edge in G that minimizes the dilation of the resulting graph. The fastest algorithm to date for this problem has O(n4) running time and uses O(n2) space. We show how to improve the running time to O(n3logn) while maintaining quadratic space requirement. In fact, our algorithm not only determines the best shortcut but computes the dilation of G∪{(u,v)} for every pair of distinct vertices u and v.