When crossings count - approximating the minimum spanning tree

In this paper, we present an (1+ε)-approximation algorithm to 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 is O ((n/ε⁵)α³(n) log⁵ n), where ε>0 is a prescribed constant. In the second part of our paper, we 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, GIV99] for problems involving points with the crossing metric as a distance function. Applications include MST, matching, clustering, nearest-neighbor, and furthest-neighbor.