Fast pruning of geometric spanners

Let S be a set of points in R d . Given a geometric spanner graph, G = (S,E), with constant stretch factor t, and a positive constant e, we show how to construct a (1+e)-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 O(|E| log |S|). We obtain our result by designing a new algorithm that finds the pair in a well-separated pair decomposition separating two given query points. Previously, it was known how to answer such a query in O(log|S|) time. We show how a sequence of such queries can be answered in O(1) amortized time per query, provided all query pairs are from a polynomially bounded range