Optimal Las Vegas Approximate Near Neighbors in ℓ p

Abstract

We show that approximate near neighbor search in high dimensions can be solved in a Las Vegas fashion (i.e., without false negatives) for ℓ p (1≤ p ≤ 2) while matching the performance of optimal locality-sensitive hashing. Specifically, we construct a data-independent Las Vegas data structure with query time O ( dn ρ ) and space usage O ( dn 1+ρ ) for ( r, c r )-approximate near neighbors in R d under the ℓ p norm, where ρ = 1/ c p + o (1). Furthermore, we give a Las Vegas locality-sensitive filter construction for the unit sphere that can be used with the data-dependent data structure of Andoni et al. (SODA 2017) to achieve optimal space-time tradeoffs in the data-dependent setting. For the symmetric case, this gives us a data-dependent Las Vegas data structure with query time O ( dn ρ ) and space usage O ( dn 1+ρ ) for ( r, c r )-approximate near neighbors in R d under the ℓ p norm, where ρ = 1/(2 c p - 1) + o (1). Our data-independent construction improves on the recent Las Vegas data structure of Ahle (FOCS 2017) for ℓ p when 1 < p ≤ 2. Our data-dependent construction performs even better for ℓ p for all pε [1, 2] and is the first Las Vegas approximate near neighbors data structure to make use of data-dependent approaches. We also answer open questions of Indyk (SODA 2000), Pagh (SODA 2016), and Ahle by showing that for approximate near neighbors, Las Vegas data structures can match state-of-the-art Monte Carlo data structures in performance for both the data-independent and data-dependent settings and across space-time tradeoffs.