Fast Nearest Neighbor Search for $\ell_p$ Metrics

2026-03-22

AI Summary

This work addresses the challenge of efficient approximate nearest neighbor search in $\ell_p$ metric spaces for $p > 2$ under large-scale data settings. The authors propose a novel approach combining randomized data structures with metric embeddings, leveraging a careful analysis of the geometric properties of $\ell_p$ spaces to design an effective indexing strategy. Their method achieves query time nearly logarithmic in the dataset size—specifically $\text{poly}(d \log n)$—with only polynomial space overhead $\text{poly}(dn)$ and provably controlled approximation error. A key contribution is the first known approximation ratio of $p^{O(1) + \log \log p}$, which significantly improves upon or is incomparable to the best prior results (e.g., from 2019 and 2025), thereby overcoming a longstanding performance bottleneck in high-dimensional $\ell_p$ approximate nearest neighbor search.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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].

Problem

• Nearest Neighbor Search
• ℓ_p metrics
• fast query time
• approximation
• high-dimensional spaces

Innovation

• Nearest Neighbor Search
• ℓ_p metrics
• fast query time
• randomized data structure
• approximation algorithms