Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say Õ(n) bits. There is an inherent tradeoff between the sparsity parameter δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when δ ≥ 1/3 there are schemes with constant additive stretch (distance d is stretched to… CONTINUE READING