\begin{algorithm}[H]
    \centering
    \small
    \caption{SNN Index}\label{algo:index}
    \begin{algorithmic}[1]
    \State \textbf{Input:} Data matrix $P=[p_1,p_2,\ldots,p_n]^T \in \mathbb{R}^{n \times d}$
    \State Compute $\mu := \mathrm{mean}(\{p_j\})$
    \State Compute the mean-centered matrix $X$ with rows $x_i:= p_i - \mu$
    \State Compute the singular value decomposition of $X=U\Sigma V^T$
    \State Compute the sorting keys $\alpha_i = x_i^T v_1$ for  $i=1,2,\ldots,n$
    \State Sort data points $X$ such that $\alpha_1\leq \alpha_2\leq \cdots\leq \alpha_n$
    \State Compute $\overline{x_i} = (x_i^T x_i)/2$ for $i=1,2,\ldots,n$
    \State \textbf{Return:} $\mu$, $X$, $v_1$, $[\alpha_i]$, $[\overline{x_i}]$
    \end{algorithmic}
\end{algorithm}

\begin{algorithm}
[H]
\centering
\small
\caption{SNN Index} \begin{algorithmic}
[1]
\State \textbf{Input:} Data matrix $P=[p_1,p_2,\ldots,p_n]^T \in \mathbb{R}^{n \times d}$
\State Compute $\mu := \mathrm{mean}(\{p_j\})$
\State Compute the mean-centered matrix $X$ with rows $x_i:= p_i - \mu$
\State Compute the singular value decomposition of $X=U\Sigma V^T$
\State Compute the sorting keys $\alpha_i = x_i^T v_1$ for $i=1,2,\ldots,n$
\State Sort data points $X$ such that $\alpha_1\leq \alpha_2\leq \cdots\leq \alpha_n$
\State Compute $\overline{x_i} = (x_i^T x_i)/2$ for $i=1,2,\ldots,n$
\State \textbf{Return:} $\mu$, $X$, $v_1$, $[\alpha_i]$, $[\overline{x_i}]$
\end{algorithmic}
\end{algorithm}
{
    "title": "fast and exact fixed-radius neighbor search based on sorting",
    "id": "2212.07679",
    "abstract": "fixed-radius near neighbor search is a fundamental data operation that retrieves all data points within a user-specified distance to a query point. there are efficient algorithms that can provide fast approximate query responses, but they often have a very compute-intensive indexing phase and require careful parameter tuning. therefore, exact brute force and tree-based search methods are still widely used. here we propose a new fixed-radius near neighbor search method, called snn, that significantly improves over brute force and tree-based methods in terms of index and query time, provably returns exact results, and requires no parameter tuning. snn exploits a sorting of the data points by their first principal component to prune the query search space. further speedup is gained from an efficient implementation using high-level basic linear algebra subprograms (blas). we provide theoretical analysis of our method and demonstrate its practical performance when used stand-alone and when applied within the dbscan clustering algorithm.",
    "categories": "cs.ir cs.ds",
    "doi": "",
    "created": "2022-12-15",
    "updated": "2024-01-29",
    "authors": [
        "xinye chen",
        "stefan g\u00fcttel"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2212.07679"
}
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