    \begin{algorithmic}[1]
    \State \textbf{Input:} Query vector $q$; user-specified radius $R$; output from Algorithm~\ref{algo:index}
    \State Compute $x_q: = q - \mu$
    \State Compute the sorting score of $x_q$, i.e., $\alpha_q := x_q^T v_1$
    \State Select candidate index range $J$ so that $|\alpha_j - \alpha_q| \le R$ for all $j\in J$
    \State Compute $d := \overline{x}(J) - X(J,:)^T x_q$ using the precomputed $ \overline{x} = [\overline{x_i}]$
    \State \textbf{Return:} Points $x_j$ with $d_j \leq (R^2 - x_q^T x_q)/2$ according to \eqref{eq:ip2}
    \end{algorithmic}

\begin{algorithmic}
[1]
\State \textbf{Input:} Query vector $q$; user-specified radius $R$; output from Algorithm~\ref{algo:index}
\State Compute $x_q: = q - \mu$
\State Compute the sorting score of $x_q$, i.e., $\alpha_q := x_q^T v_1$
\State Select candidate index range $J$ so that $|\alpha_j - \alpha_q| \le R$ for all $j\in J$
\State Compute $d := \overline{x}(J) - X(J,:)^T x_q$ using the precomputed $ \overline{x} = [\overline{x_i}]$
\State \textbf{Return:} Points $x_j$ with $d_j \leq (R^2 - x_q^T x_q)/2$ according to \eqref{eq:ip2}
\end{algorithmic}
{
    "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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