Optimal Las Vegas Approximate Near Neighbors in $\ell_p$

TL;DR

This paper introduces Las Vegas algorithms for approximate near neighbor search in high-dimensional $ ext{L}_p$ spaces, achieving performance comparable to the best randomized methods without false negatives, and improves upon previous data structures.

Contribution

It presents the first Las Vegas data structures for approximate near neighbors in $ ext{L}_p$ spaces, matching the performance of optimal randomized algorithms and incorporating data-dependent techniques.

Findings

Achieves $O(dn^{ ho})$ query time with $O(dn^{1+ ho})$ space for $ ext{L}_p$ norms.

Introduces data-independent and data-dependent Las Vegas data structures.

Matches performance of state-of-the-art Monte Carlo methods in the Las Vegas setting.

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 $\ell_p$ ($1\le p\le 2$) 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.