Aren't we all nearest neighbors: Spatial trees, high dimensional reductions and batch nearest neighbor search

Parallel Nearest Neighbors in Low Dimensions with Batch Updates

We present a set of parallel algorithms for computing exact k-nearest neighbors in low dimensions. Many k-nearest neighbor algorithms use either a kd-tree or the Morton ordering of the point set; our algorithms combine these approaches using a data structure we call the \textit{zd-tree}. We show that this combination is both theoretically efficient under common assumptions, and fast in practice. For point sets of size $n$ with bounded expansion constant and bounded ratio, the zd-tree can be built in $O(n)$ work with $O(n^ε)$ span for constant $ε<1$, and searching for the $k$-nearest neighbors …

通过近似最近邻搜索实现快速几何稀疏生成器

Fast Geometric Spanners via Approximate Nearest Neighbor Search

本文利用批量近似最近邻搜索黑盒，在一般度量空间中实现了次二次时间构造低失真稀疏生成器，并应用于快速近似Wasserstein距离，首次在$\ell_p$空间（$p>2$）取得突破。

The nearest-colattice algorithm

In this work, we exhibit a hierarchy of polynomial time algorithms solving approximate variants of the Closest Vector Problem (CVP). Our first contribution is a heuristic algorithm achieving the same distance tradeoff as HSVP algorithms, namely $\approx β^{\frac{n}{2β}}\textrm{covol}(Λ)^{\frac{1}{n}}$ for a random lattice $Λ$ of rank $n$. Compared to the so-called Kannan's embedding technique, our algorithm allows using precomputations and can be used for efficient batch CVP instances. This implies that some attacks on lattice-based signatures lead to very cheap forgeries, after a precomputati…

Improved approximate near neighbor search without false negatives for $l_2$

We present a new algorithm for the $c$--approximate nearest neighbor search without false negatives for $l_2^d$. We enhance the dimension reduction method presented in \cite{wygos_red} and combine it with the standard results of Indyk and Motwani~\cite{motwani}. We present an efficient algorithm with Las Vegas guaranties for any $c>1$. This improves over the previous results, which require $c=ω(\log\log{n})$ \cite{wygos_red}, where $n$ is the number of the input points. Moreover, we improve both the query time and the pre-processing time. Our algorithm is tunable, which allows for different co…