Sublinear-time approximation of Euclidean minimum spanning tree

We consider the problem of finding the weight of a Euclidean minimum spanning tree for a set of n points in ℝ^d. We focus on the situation when the input point set is supported by certain basic (and commonly used) geometric data structures that can provide efficient access to the input in a structured way. We present an algorithm that estimates with high probability the weight of a Euclidean minimum spanning tree of a set of points to within 1 + ε using only Õ(√n poly(1/ε)) queries for constant d. The algorithm assumes that the input is supported by a minimal bounding cube enclosing it, by orthogonal range queries, and by cone approximate nearest neighbors queries.