An Optimal Localized Approximation Scheme for Euclidean MST

We consider the problem of locally constructing a spanning subgraph that approximates the Euclidean minimum spanning tree of a unit disk graph. We show that for any k ≥ 2 there exists a klocalized distributed algorithm that, given a unit disk graph U in the plane, constructs a planar subgraph of U containing a Euclidean MST on V (U), whose degree is at most 6, and whose total weight is at most 1+ 2 k−1 times the weight of the Euclidean MST on V (U). We prove that this approximation bound is tight by showing that, for any > 0, no k-localized algorithm can construct a spanning subgraph of U whose total weight is less than 1+ 2 k−1− times the weight of a Euclidean MST on V (U). En route to our results, we prove a nice result about weighted planar graphs.