Efficient Construction of a Bounded Degree Spanner with Low Weight

Let S be a set of n points in ℝd 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 logdn). Applying recent results of Das, Narasimhan and Salowe to this t-spanner gives an O(n logdn) 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(n2) 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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