We study the following problem: preprocess a set $$\mathcal {O}$$O of objects into a data structure that allows us to efficiently report all pairs of objects from $$\mathcal {O}$$O that intersect inside an axis-aligned query range $${Q}$$Q. We present data structures of size $$O(n\cdot {{\mathrm{polylog\,}}}n)$$O(n·polylogn) and with query time $$O((k+1)\cdot {{\mathrm{polylog\,}}}n)$$O((k+1)·polylogn) time, where k is the number of reported pairs, for two classes of objects in $${\mathbb R}^2$$R2: axis-aligned rectangles and objects with small union complexity. For the 3-dimensional case where the objects and the query range are axis-aligned boxes in $${\mathbb R}^3$$R3, we present a data structure of size $$O(n\sqrt{n}\cdot {{\mathrm{polylog\,}}}n)$$O(nn·polylogn) and query time $$O((\sqrt{n}+k)\cdot {{\mathrm{polylog\,}}}n)$$O((n+k)·polylogn). When the objects and query are fat, we obtain $$O((k+1)\cdot {{\mathrm{polylog\,}}}n)$$O((k+1)·polylogn) query time using $$O(n\cdot {{\mathrm{polylog\,}}}n)$$O(n·polylogn) storage.

中文翻译：

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在查询范围内查找成对交点

我们研究以下问题：将一组 $$\mathcal {O}$$O 的对象预处理为一个数据结构，该数据结构允许我们有效地报告 $$\mathcal {O}$$O 中相交的所有对象对轴对齐查询范围 $${Q}$$Q。我们提出了大小为 $$O(n\cdot {{\mathrm{polylog\,}}}n)$$O(n·polylogn) 和查询时间 $$O((k+1)\cdot { {\mathrm{polylog\,}}}n)$$O((k+1)·polylogn) 时间，其中 k 是报告对的数量，对于 $${\mathbb R}^2 中的两类对象$$R2：轴对齐的矩形和联合复杂度小的对象。对于对象和查询范围是 $${\mathbb R}^3$$R3 中的轴对齐框的 3 维情况，我们提出了一个大小为 $$O(n\sqrt{n}\ cdot {{\mathrm{polylog\,}}}n)$$O(nn·polylogn) 和查询时间 $$O((\sqrt{n}+k)\cdot {{\mathrm{polylog\,}} }n)$$O((n+k)·polylogn)。