Optimal Las Vegas Approximate Near Neighbors In \(\ell_p\)
Alexander Wei . Arxiv 2018 – 0 citations
[Paper]
We show that approximate near neighbor search in high dimensions can be solved in a Las Vegas fashion (i.e., without false negatives) for (\ell_p) ((1\le p\le 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 usage (O(dn^{1+\rho})) for ((r, c r))-approximate near neighbors in (\mathbb{R}^{d}) under the (\ell_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, c r))-approximate near neighbors in (\mathbb{R}^{d}) under the (\ell_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 (\ell_p) when (1 < p\le 2). Our data-dependent construction does even better for (\ell_p) for all (p\in [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.