Lower bounds on the distortion of embedding finite metric spaces in graphs

The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions. The existing constructions of graph spanners imply that any n-point metric space can be represented by a (weighted) graph with n vertices and n^1+0(1/r) edges, with distances distorted by at most r. We show that this tradeoff between the number of edges and the distortion cannot be improved, and that it holds in a much more general setting. The main technical lemma claims that the metric space induced by an unweighted graph H of girth g cannot be embedded in a graph G (even if it is weighted) of smaller Euler characteristic, with distortion less than g/4 - 3/2. In the special case when | V(G)\ = | V(H)| and G has strictly less edges than H, an improved bound of g/3 - 1 is shown. In addition, we discuss the case χ(G) < χ(H) - 1, as well as some interesting higher-dimensional analogues. The proofs employ basic techniques of algebraic topology.