Fast Geometric Spanners via Approximate Nearest Neighbor Search

Abstract

We study the problem of constructing metric spanners in general metric spaces in subquadratic time when given blackbox access to a fast algorithm for batch approximate nearest neighbor search. In particular, we show the following results for any metric space $\mathsf{M} = ([n], \mathsf{d})$ with aspect ratio $Δ$ admitting a $c$-approximate batch nearest neighbor search algorithm with runtime $τ_{\mathsf{M}}(n)$, (1) There exists an algorithm that, for any $k \in \mathbb{N}$, constructs an $O(c k)$-distortion spanner with $\tilde{O}(kn^{1+1/2k} \log Δ)$ edges and runs in time $\tilde{O}(τ_{\mathsf{M}} \cdot k n^{1/k} \log Δ)$. (2) Any algorithm that learns at most $o(n^{1+1/k}/k)$ pairwise distances by querying a distance oracle and a blackbox batch nearest neighbor search oracle necessarily incurs $Ω(c k)$ distortion. Our results entail that (truly) sub-quadratic time algorithms for spanner construction is equivalent to subquadratic time BANN (up to constant-factor losses). As a further application, we use our fast spanner constructions to obtain a fast algorithm for approximating the Wasserstein distance $\mathsf{W}_q$, for all $q > 1$, over any metric space admitting an efficient batch approximate nearest neighbor search algorithm. Together with recent new efficient algorithms for approximate nearest neighbor search in $\ell_p$ spaces, for $p > 2$, our results entail the first subquadratic time algorithms for spanner construction (with the stated size-distortion tradeoff) and $\mathsf{W}_q$ distance approximation over these metric spaces.