Fast Nearest Neighbor Search for $\ell_p$ Metrics
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912977569972224 |
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| author | Krauthgamer, Robert Petruschka, Nir |
| author_facet | Krauthgamer, Robert Petruschka, Nir |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21148 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast Nearest Neighbor Search for $\ell_p$ Metrics Krauthgamer, Robert Petruschka, Nir Data Structures and Algorithms Computational Geometry Metric Geometry 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]. |
| title | Fast Nearest Neighbor Search for $\ell_p$ Metrics |
| topic | Data Structures and Algorithms Computational Geometry Metric Geometry |
| url | https://arxiv.org/abs/2603.21148 |