Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are between any two successive hyperplanes. Let $|A(S,m)|$ be the number of different closest pairs in the ${{m+1} \choose 2}$ vertical slabs that are bounded by $H_i$ and $H_j$, over all $1 \leq i < j \leq m+1$. We prove tight bounds for the largest possible value of $|A(S,m)|$, over all point sets of size $n$, and for all values of $1 \leq m \leq n$. As a result of these bounds, we obtain, for any constant $\epsilon>0$, a data structure of size $O(n)$, such that for any vertical query slab $Q$, the closest pair in the set $Q \cap S$ can be reported in $O(n^{1/2+\epsilon})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

翻译：设 $S$ 为 $\mathbb{R}^d$ 中 $n$ 个点的集合，其中 $d \geq 2$ 为常数，并设 $H_1,H_2,\ldots,H_{m+1}$ 为一列按第一坐标排序的垂直超平面，使得任意两个相邻超平面之间恰好包含 $S$ 中的 $n/m$ 个点。令 $|A(S,m)|$ 表示在所有 $1 \leq i < j \leq m+1$ 的范围内，由 $H_i$ 和 $H_j$ 界定的 ${{m+1} \choose 2}$ 个垂直条带中不同最邻近点对的数量。我们证明了对于所有大小为 $n$ 的点集以及所有 $1 \leq m \leq n$ 的取值，$|A(S,m)|$ 可能的最大值的紧界。基于这些界，我们针对任意常数 $\epsilon>0$，构建了一个大小为 $O(n)$ 的数据结构，使得对于任意垂直查询条带 $Q$，可在 $O(n^{1/2+\epsilon})$ 时间内报告集合 $Q \cap S$ 中的最邻近点对。在此工作之前，尚未存在具有亚线性查询时间的线性空间数据结构。