Euclidean spanners in high dimensions

A classical result in metric geometry asserts that any n-point metric admits a linear-size spanner of dilation O(log n) [PS89]. More generally, for any c > 1, any metric space admits a spanner of size O(n), and dilation at most c. This bound is tight assuming the well-known girth conjecture of Erdős [Erd63]. We show that for a metric induced by a set of n points in high-dimensional Euclidean space, it is possible to obtain improved dilation/size trade-offs. More specifically, we show that any n-point Euclidean metric admits a near-linear size spanner of dilation O( √ log n). Using the LSH scheme of Andoni and Indyk [AI06] we further show that for any c > 1, there exist spanners of size roughly O(n 2 ) and dilation O(c). Finally, we also exhibit super-linear lower bounds on the size of spanners with constant dilation.