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 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(dn ρ ) and space usage O(dn 1+ρ ) for (r, cr)-approximate near neighbors in R d under the ℓp 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, cr)-approximate near neighbors in R d under the ℓp norm, where ρ = 1/(2c p − 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 does even better for ℓp for all p ∈ [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

Faster Deterministic and Las Vegas Algorithms for Offline Approximate Nearest Neighbors in High Dimensions

“…Not only is the time bound better than LSH for ε sufficiently small, but the approach is also less involved than the previous Las-Vegas-ification approaches for LSH [38,2,43]. Essentially, we show that the simple idea of using random partitions instead of random samples, as first suggested by Indyk [27] (and also used in part in subsequent methods [38,2,43]), is compatible with the polynomial method from [5], after some technical modifications.…”

“…A fundamental question is whether these techniques can be efficiently derandomized. Finding Las Vegas randomized algorithms with comparable performance is already a nontrivial problem, and has been the subject of several recent papers [38,2,43]. Deterministic algorithms seem even more challenging.…”

“…If the goal is to just eliminate errors in the output, better results are possible with randomized Las Vegas algorithms. As mentioned, there were a series of papers on turning LSH into Las Vegas algorithms by Pagh (SODA 2016) [38], Ahle (FOCS 2017) [2], and Wei (SODA 2019) [43], but any (dataoblivious or data-dependent) LSH-based method requires at least n 2−Θ(ε) time [34,37] to answer n queries.…”

On Adaptive Distance Estimation

“…The Locality Sensitive Hashing (LSH) approach of [IM98] gives a Monte Carlo randomized approach with low memory and query time, but it does not support adaptive queries. There has also been recent interest in obtaining Las Vegas versions of such algorithms [Ahl17,Wei19,Pag18,SW17]. Unfortunately, those works also do not support adaptive queries.…”