Before revision:  This property is desirable in machine learning . In this paper, we show that this metricis equal to previously studied geodesic-based metric defined on points in Euclidean space. Previous best work showed that these metrics are 3-approximations of each other. It was not known or suspected that these metrics were equal. 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.
After revision:  This property is desirable in machine learning , or it can be seen as an example of a manifold metric. Previous computational research on such metrics despaired of computing exact distances on account of the apparent difficulty of minimizing over all continuous paths between a pair of points. We leverage the exact computation of the nearest neighbor metric 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.