Optimal Las Vegas Approximate Near Neighbors in ℓp

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 ℓp (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(dnρ) and space usage O(dn1+ρ) for (r, c r)-approximate near neighbors in Rd under the ℓp norm, where ρ = 1/cp + 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(dnρ) and space usage O(dn1+ρ) for (r, c r)-approximate near neighbors in Rd under the ℓp norm, where ρ = 1/(2cp - 1) + o(1). Our data-independent construction improves on the recent Las Vegas data structure of Ahle (FOCS 2017) for ℓp when 1 < p≤ 2. Our data-dependent construction performs even better for ℓp 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.