The {\em bottleneck distance} is a natural measure of the distance between two finite point sets of equal cardinality, defined as the minimum over all bijections between the point sets of the maximum distance between any pair of points put in correspondence by the bijection. In this work, we consider the problem of building a data structure $\mathbb{D}$ that indexes a collection of $m$ planar point sets (of varying sizes) and supports nearest bottleneck distance queries: given a query point set $Q$ of size $n$, we would like to find the point set(s) $P \in \mathbb{D}$ of size $n$ that are closest in terms of bottleneck distance. Without loss of generality, we assume that all point sets belong to the unit box $[0,1]^2$ in the plane and focus on the $L_\infty$ norm, although the techniques can also be used for other norms. The main contribution is a {\em trie}-based data structure finds a $6$-approximate nearest neighbor in $O(-\lg(d_B(\mathbb{D},Q)) n)$ time, where $d_B(\mathbb{D},Q)$ is the minimum bottleneck distance from $Q$ to any point set in $\mathbb{D}$.

翻译：{em clockneck 距离} 是两组相同基点的固定点之间的距离的自然度量, 定义为两组点数之间的最小点数, 每组点数之间最大距离的最小值值。 在这项工作中, 我们考虑建立数据结构$\ mathbb{D} 美元的问题, 该数据结构将汇集一组( 不同大小的) 美元平面点数, 支持最接近的瓶颈距离查询: 给一个查询点数, 定额为 $Q 美元, 大小为 美元= mathb{D}, 大小为 美元, 大小为 美元, 大小为 美元=xxxxxx 。 我们假设所有点数都属于 $0, 1\2美元在平面上, 重点是 $L\ intyfty 规范, 尽管技术也可以用于其他规范。 基于数据结构的主要捐款是 $D$( b_ b_ b) 至 nxx n_ b) 任何时间的 6美元近 Q (- d_ b) b) 美元, 。