# Efficient construction of a bounded degree spanner with low weight

## Abstract

Let *S* be a set of *n* points in ℝ^{d} and let *t*>1 be a real number. A *t*-spanner for *S* is a graph having the points of *S* as its vertices such that for any pair *p, q* of points there is a path between them of length at most *t* times the Euclidean distance between *p* and *q*. An efficient implementation of a greedy algorithm is given that constructs a *t*-spanner having bounded degree such that the total length of all its edges is bounded by *O*(log *n*) times the length of a minimum spanning tree for *S*. The algorithm has running time *O(n* log^{d}*n*). Applying recent results of Das, Narasimhan and Salowe to this *t*-spanner gives an *O(n* log^{d}*n*) time algorithm for constructing a *t*-spanner having bounded degree and whose total edge length is proportional to the length of a minimum spanning tree for *S*. Previously, no *o(n*^{2}) time algorithms were known for constructing a *t*-spanner of bounded degree. In the final part of the paper, an application to the problem of distance enumeration is given.

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