Before revision:  In this paper, we prove that the edge-squared and nearest-neighbor metrics are in fact equivalent . Previous best work showed that  the  edge-squared metric was a 3-approximation of the Nearest Neighbor metric.
After revision:  In this paper, we prove that the edge-squared and nearest-neighbor metrics are in fact equivalent . 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 edge-squared metric was a 3-approximation of the Nearest Neighbor metric.