Given a connected geometric graph G, we consider the problem of constructing a t-spanner of G having the minimum number of edges. We prove that for every t with 1 1+1/t) edges. This bound almost matches the known upper bound, which states that every connected weighted graph with n vertices contains a t-spanner with O(tn1+2/(t+1)) edges. We also prove that the problem of deciding whether a given geometric graph contains a t-spanner with at most K edges is NP-hard. Previously, this NP-hardness result was only known for non-geometric graphs.
Let (S,d) be a finite metric space, where each element p S has a non-negative weight w(p). We study spanners for the set S with respect to weighted distance function d w , where d w (p,q) is w(p)+d(p,q)+wq if p≠q and 0 otherwise. We present a general method for turning spanners with respect to the d-metric into spanners with respect to the d w -metric. For any given ε>0, we can apply our method to obtain (5+ε)-spanners with a linear number of edges for three cases: points in Euclidean space ℝ d , points in spaces of bounded doubling dimension, and points on the boundary of a convex body in ℝ d where d is the geodesic distance function. We also describe an alternative method that leads to (2+ε)-spanners for points in ℝ d and for points on the boundary of a convex body in ℝ d . The number of edges in these spanners is O(nlogn). This bound on the stretch factor is nearly optimal: in any finite metric space and for any ε>0, it is possible to assign weights to the elements such that any non-complete graph has stretch factor larger than 2-ε.