  Several researchers proposed using non-Euclidean metrics on point sets in
Euclidean space for clustering noisy data. Almost always, a distance function
is desired that recognizes the closeness of the points in the same cluster,
even if the Euclidean cluster diameter is large. Therefore, it is preferred to
assign smaller costs to the paths that stay close to the input points.
  In this paper, we consider the most natural metric with this property, which
we call the nearest neighbor metric. Given a point set P and a path $\gamma$,
our metric charges each point of $\gamma$ with its distance to P. The total
charge along $\gamma$ determines its nearest neighbor length, which is formally
defined as the integral of the distance to the input points along the curve. We
describe a $(3+\varepsilon)$-approximation algorithm and a
$(1+\varepsilon)$-approximation algorithm to compute the nearest neighbor
metric. Both approximation algorithms work in near-linear time. The former uses
shortest paths on a sparse graph using only the input points. The latter uses a
sparse sample of the ambient space, to find good approximate geodesic paths.
