This paper considers enumerating answers to similarity-join queries under dynamic updates: Given two sets of $n$ points $A,B$ in $\mathbb{R}^d$, a metric $\phi(\cdot)$, and a distance threshold $r > 0$, report all pairs of points $(a, b) \in A \times B$ with $\phi(a,b) \le r$. Our goal is to store $A,B$ into a dynamic data structure that, whenever asked, can enumerate all result pairs with worst-case delay guarantee, i.e., the time between enumerating two consecutive pairs is bounded. Furthermore, the data structure can be efficiently updated when a point is inserted into or deleted from $A$ or $B$. We propose several efficient data structures for answering similarity-join queries in low dimension. For exact enumeration of similarity join, we present near-linear-size data structures for $\ell_1, \ell_\infty$ metrics with $\log^{O(1)} n$ update time and delay. We show that such a data structure is not feasible for the $\ell_2$ metric for $d \ge 4$. For approximate enumeration of similarity join, where the distance threshold is a soft constraint, we obtain a unified linear-size data structure for $\ell_p$ metric, with $\log^{O(1)} n$ delay and update time. In high dimensions, we present an efficient data structure with worst-case delay-guarantee using locality sensitive hashing (LSH).

中文翻译：

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相似联接的动态枚举

本文考虑枚举动态更新下的相似联接查询的答案：给定$ \ mathbb {R} ^ d $中的两个$ n $点$ A，B $，度量$ \ phi（\ cdot）$，以及距离阈值$ r> 0 $，报告所有对点$（a，b）\在A \时间B $中具有$ \ phi（a，b）\ le r $。我们的目标是将$ A，B $存储到动态数据结构中，每当被问到时，它都可以枚举具有最坏情况延迟保证的所有结果对，即，枚举两个连续对之间的时间是有界的。此外，当将点插入到$ A $或$ B $中或从$ A $或$ B $中删除时，可以有效地更新数据结构。我们提出了几种有效的数据结构，用于在低维情况下回答相似性联接查询。为了精确枚举相似性联接，我们给出了$ \ ell_1的近似线性大小的数据结构，\ ell_ \ infty $指标，其中$ \ log ^ {O（1）} n $更新时间和延迟。我们表明，这种数据结构对于$ d \ ge 4 $的$ \ ell_2 $度量标准是不可行的。对于相似连接的近似枚举，其中距离阈值是一个软约束，我们为$ \ ell_p $度量获取统一的线性大小数据结构，并具有$ \ log ^ {O（1）} n $延迟和更新时间。在高维度上，我们使用局部敏感哈希（LSH）提出了一种有效的数据结构，具有最坏情况的延迟保证。