Sub-quadratic (1+ϵ)-approximate Euclidean Spanners, with Applications

We study graph spanners for point-set in the high-dimensional Euclidean space. On the one hand, we prove that spanners with stretch $\lt \sqrt{2}$ and subquadratic size are not possible, even if we add Steiner points. On the other hand, if we add extra nodes to the graph (non-metric Steiner points), then we can obtain $(1+\epsilon)$-approximate spanners of subquadratic size. We show how to construct a spanner of size $n^{2-\Omega\left(\epsilon^{3}\right)}$, as well as a directed version of the spanner of size $n^{2-\Omega\left(\epsilon^{2}\right)}$. We use our directed spanner to obtain an algorithm for computing $(1+\epsilon)$-approximation to Earth-Mover Distance (optimal transport) between two sets of size n in time $n^{2-\Omega\left(\epsilon^{2}\right)}$.