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Sub-quadratic (1+\eps)-approximate Euclidean Spanners, with Applications (2310.05315v1)

Published 9 Oct 2023 in cs.CG and cs.DS

Abstract: We study graph spanners for point-set in the high-dimensional Euclidean space. On the one hand, we prove that spanners with stretch <\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+\eps)-approximate spanners of subquadratic size. We show how to construct a spanner of size n{2-\Omega(\eps3)}, as well as a directed version of the spanner of size n{2-\Omega(\eps2)}. We use our directed spanner to obtain an algorithm for computing (1+\eps)-approximation to Earth-Mover Distance (optimal transport) between two sets of size n in time n{2-\Omega(\eps2)}.

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