Note on Bounded Degree Spanners for Doubling Metrics

We focus on obtaining sparse representations of metrics: these are called spanners, and they have been studied extensively both for general and Euclidean metrics. Formally, a t-spanner for a metric M = (V, d) is an undirected graph G = (V,E) such that the distances according to dG (the shortest-path metric of G) are close to the distances in d: i.e., d(u, v) ≤ dG(u, v) ≤ t d(u, v). Clearly, one can take a complete graph and obtain t = 1, and hence the quality of the spanner is typically measured by how few edges can G contain whilst maintaining a stretch of at most t. The notion of spanners has been widely studied for general metrics (see, e.g. [PS89, ADD+93, CDNS95]), and for geometric distances (see, e.g., [CK95, Sal91, Vai91, ADM+95]). Here, we are particularly interested in the case when the input metric has bounded doubling dimension and the spanner we want to construct has small stretch, i.e. t = 1 + ε, for small ε > 0. We show that for fixed ε and metrics with bounded doubling dimension, it is possible to construct linear sized (1 + ε)-spanners. Observe that any 1.5-spanner for a uniform metric on n points must be the complete graph. Hence, without any restriction on the input metric, it is not possible to construct an (1 + ε)-spanner with linear number of edges.