Computing the dilation of edge-augmented graphs in metric spaces

Abstract

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 $O(n^4)$ running time and uses $O(n^2)$ space. We show how to improve the running time to $O(n^3\log n)$ while maintaining quadratic space requirement. In fact, our algorithm not only determines the best shortcut but computes the dilation of $G\cup \{(u,v)\}$ for every pair of distinct vertices u and v.