Fast Nearest Neighbor Search for $\ell_p$ Metrics

TL;DR

This paper introduces a randomized data structure for approximate nearest neighbor search in  spaces with p>2, achieving fast query times and improved approximation guarantees suitable for large-scale applications.

Contribution

It presents a novel data structure for  NNS in  spaces with p>2, offering faster query times and competitive approximation ratios compared to previous methods.

Findings

Achieves  NNS with p>2 in poly(dn) space.

Provides approximation factor of p^{O(1)+\u221a{\u2212}log p}.

Outperforms or matches state-of-the-art in fast query regimes.

Abstract

The Nearest Neighbor Search (NNS) problem asks to design a data structure that preprocesses an $n$-point dataset $X$ lying in a metric space $\mathcal{M}$, so that given a query point $q \in \mathcal{M}$, one can quickly return a point of $X$ minimizing the distance to $q$. The efficiency of such a data structure is evaluated primarily by the amount of space it uses and the time required to answer a query. We focus on the fast query-time regime, which is crucial for modern large-scale applications, where datasets are massive and queries must be processed online, and is often modeled by query time $\text{poly}(d \log n)$. Our main result is such a randomized data structure for NNS in $\ell_p$ spaces, $p>2$, that achieves $p^{O(1) + \log\log p}$ approximation with fast query time and $\text{poly}(dn)$ space. Our data structure improves, or is incomparable to, the state-of-the-art for the fast query-time regime from [Bartal and Gottlieb, TCS 2019] and [Krauthgamer, Petruschka and Sapir, FOCS 2025].