Finding the best shortcut in a geometric network

Given a Euclidean graph G in R d with n vertices and m edges we consider the problem of adding a shortcut such that the stretch factor of the resulting graph is minimized. Currently, the fastest algorithm for computing the stretch factor of a Euclidean graph runs in O(mn + n 2 log n) time, resulting in a trivial O(mn 3 + n 4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n 4) time using O(n 2) space. To reduce the running times we consider several approximation algorithms. Our main result is a(2+ε)-approximation algorithm with running time O(nm + n 2 (log n +1/ε 3d)) using O(n 2) space.