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 ℝd. 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(√ 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.

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