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. This paper represents one of the first proofs of equating a continuous metric with a discrete metric, using non-trivial discrete methods. Our proof uses the Kirszbraun theorem (also known as the Lipschitz Extension Theorem and Brehm's Extension Theorem), a notable theorem in functional analysis and computational geometry. The results of our paper, combined with the results of Hwang, Damelin, and Hero, tell us that the Nearest Neighbor distance on i. i.d samples of a density is a reasonable constant approximation of a natural density-based distance function .
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 edge-squared metric are faster to compute and sparser than the best known Euclidean spanners in a variety of settings .