Approximating the weight of the Euclidean minimum spanning tree in sublinear time

We consider the problem of computing the weight of a Euclidean minimum spanning tree for a set of n points in ℝ ^d. We focus on the setting where 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 neighbor queries.