Let S be a set of points in R. Given a geometric spanner graph, G = (S, E), with constant dilation t, and a positive constant ε, we show how to construct a (1 + ε)-spanner of G with O(|S|) edges in time O(|E|+ |S| log |S|). Previous algorithms require a preliminary step in which the edges are sorted in non-decreasing order of their lengths and, thus, have running time Ω(|E| log |S|). We obtain ...