Approximating the Weight of the Euclidean Minimum Spanning Tree in Sublinear Time

Abstract

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+epsilon using only (O) over tilde(root n poly(1/epsilon)) 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.