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 the Euclidean MST problem. Spanners of the edge-squared metric was a 3-approximation of the Nearest Neighbor metric.