Before revision:  We give fast algorithms to compute sparse spanners of this distance, a problem that can be seen as a generalization of both the Euclidean spanner and the Euclidean MST problem. Spanners of the edge-squared metric are faster to compute and sparser than the best known Euclidean spanners in a variety of settings .
After revision:  We give fast algorithms to compute sparse spanners of this distance, a problem that can be seen as a generalization of both the Euclidean spanner and the metric built from point sets drawn from an underlying distribution and consider the more general case of inputs that are finite collections of path-connected compact sets. The main results connect several classical theories such as the conformal change of Riemannian metrics, the theory of positive definite functions of Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop novel proof techniques based on the combination of screw functions and Lipschitz extensions that may be of independent interest .