Let V be a set of n points in 3-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any u and v in V, the length of the shortest path from u to v in the spanner is at most t times d (u, v). We show that for any t > 1, a greedy algorithm produces a t-spanner with O(n) edges, and total edge weight O(1). tot(it4ST), whereMST is a minimum spanning tree of V.

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Let V be any set of n points in k-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for any ~ and ~ in V, the length of the shortest path from u to v in the spanner is at most t times d(~, ~). We show that for any 5 > 1, there exists a polynomial-time constructihle tspanner (where ~ is a constant that depends only on 5 and k) with the following properties. Its maximum degree is 3, it has at most n 9 6 edges, and its total edge weight is comparable to the minimum spanning tree of V (for/~ < 3 its weight is O(1). wt(MgT), and for k > 3 its weight is O(log n). wt(MST)).

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