When Crossings Count — Approximating the Minimum

We present an (1+ε)-approximation algorithm for computing the minimum-spanning tree of points in a planar arrangement of lines, where the metric is the number of crossings between the spanning tree and the lines. The expected running time of the algorithm is near linear. We also show how to embed such a crossing metric of hyperplanes in d-dimensions, in subquadratic time, into high-dimensions so that the distances are preserved. As a result, we can deploy a large collection of subquadratic approximations algorithms [IM98, GIV01] for problems involving points with the crossing metric as a distance function. Applications include MST, matching, clustering, nearest-neighbor, and furthest-neighbor.