Optimal Las Vegas Approximate Near Neighbors in l(p)

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(rho)) and space usageO(dn(1+rho)) for (r, cr)-approximate near neighbors in R-d under the l(p) norm, where rho = 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(rho)) and space usage O(dn(1+rho)) for (r, cr)-approximate near neighbors in R-d under the l(p) norm, where rho = 1/(2c(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 p is an element of[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 canmatch state-of-the-art Monte Carlo data structures in performance for both the data-independent and data-dependent settings and across space-time tradeoffs.