Abstract
Given a Euclidean graph G in Rd 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+n2 log n) time, resulting in a trivial O(mn3+n4 log n) time algorithm for computing the optimal shortcut. First, we show that a simple modification yields the optimal solution in O(n4) time using O(n2) space. To reduce the running times we consider several approximation algorithms. Our main result is a (2+e)-approximation algorithm with running time O(nm+n2(log n+1/e3d)) using O(n2) space.
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