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 show that this metric is equal to previously studied geodesic-based metric defined on points in Euclidean space . Previous best work showed that the edge-squared metric was a 3-approximation of the Nearest Neighbor metric.