Sub-quadratic (1+)-approximate Euclidean Spanners, with Applications
Abstract
We study graph spanners for point-set in the high-dimensional Euclidean space. On the one hand, we prove that spanners with stretch <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+)-approximate spanners of subquadratic size. We show how to construct a spanner of size n2-(3), as well as a directed version of the spanner of size n2-(2). We use our directed spanner to obtain an algorithm for computing (1+)-approximation to Earth-Mover Distance (optimal transport) between two sets of size n in time n2-(2).
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