AI 中文总结
研究如何为向量数据库构建搜索图，引入“γ-近似可导航性”概念，证明其边数与数据集大小呈线性关系，提出随机算法近线性时间构建此类图，虽牺牲最坏情况搜索保证，但实验表明γ < 1时性能良好且更节省空间。
AI 中文摘要
基于图的方法如HNSW、DiskANN、NSG等在向量数据库中实现近似最近邻搜索越来越受欢迎。这些方法的成功促使研究如何为给定数据集构建最佳搜索图。可导航性被视为理想的图属性，与贪婪搜索结合可确保良好的近似最近邻搜索性能。然而，对于含n个向量的数据集，最稀疏的可导航图在最坏情况下需O(n√n)条边，且实验表明，对于典型的十亿节点数据集，每个节点需要数百条边，导致搜索缓慢和内存需求高。此外，在标准复杂性理论假设下，构建稀疏可导航图需Ω(n^(2 - ε))时间，对大数据集来说过高。本文引入了一种名为“γ-近似可导航性”的宽松可导航性概念，证明任何数据集都允许有仅O(n/(1 - γ))条边的γ-近似可导航图，呈数据集大小的线性关系。提出了一种随机算法在近线性时间内构建此类图。虽γ-近似可导航性牺牲了可导航性的最坏情况搜索保证，但实验表明当γ < 1时，贪婪波束搜索在此类图中仍表现良好。与完全可导航图相比，在各种数据集上获得了改进的召回率-运行时权衡。此外，我们的图更节省空间，对于可比性能，其度数通常小于完全可导航图的一半。
英文摘要
Graph-based methods like HNSW, DiskANN, NSG, and others have become an increasingly popular choice for implementing approximate nearest neighbor search (ANNS) in Vector Databases (VecDBs). The success of these methods has motivated the study of how to best construct a search graph for a given dataset. To that end, \emph{navigability} has been identified as a desirable graph property which ensures good ANNS performance when combined with greedy search. However, for a dataset with $n$ vectors, the sparsest navigable graph requires $O(n\sqrt{n})$ edges in the worst-case, and we show empirically that, for typical billion node datasets, 100s of edges are needed per node. This leads to slow search and high memory requirements. Moreover, under standard complexity theoretical assumptions, it was recently established that constructing a sparse navigable graph requires $Ω(n^{2-ε})$ time, which is prohibitive for large datasets. We address these concerns by introducing a relaxed notation of navigability called ``$γ$-almost navigability'' for any $γ\in [0,1]$, with $γ= 1$ corresponding to full navigability. We prove that any dataset (under any distance) admits a $γ$-almost navigable graph with just $O\left(\frac{n}{1-γ}\right)$ edges, linear in the dataset size. We present a randomized algorithm for constructing such a graph in near-linear time. While we prove that $γ$-almost navigability sacrifices the worst-case search guarantees enjoyed by navigability, we show empirically that greedy beam search still performs well in such graphs when $γ< 1$. Indeed, we obtain improved recall-runtime tradeoffs on a variety of datasets compared to fully navigable graphs. Moreover, our graphs are more space efficient, with degree typically less than half that of a fully navigable graph for comparable performance.
