We study the orthogonal range reporting and rectangle stabbing problems in moderate dimensions, i.e., when the dimension is $c\log (n)$ for some constant c. In orthogonal range reporting, the input is a set of n points in d dimensions, and the goal is to store these n points in a data structure such that given a query rectangle, we can report all the input points contained in the rectangle. The rectangle stabbing problem is the “dual” problem where the input is a set of rectangles, and the query is a point.

Our main result is the following: assume using $S(n)$ space, we can solve either problem in $d=c\log n$ dimensions, $c\ge 4$, using $Q(n)+O(t)$ time in the pointer machine model of computation where t is the output size. Then, we show that if the query time is small, that is, $Q(n)=n^{1-\gamma }$, for $\gamma \ge \frac{2}{2+\log c}$, then the space must be $\mathrm{\Omega }\left(n^{1-\gamma }{n}^{\sqrt{c\gamma }/e-o(\sqrt{c\gamma })}\right)$. Interestingly, we obtain this lower bound using a non-constructive method, and we show the existence of some codes that generalize a specific aspect of error correction codes. Our result overcomes the shortcomings of the previous lower bounds in the pointer machine model for non-constant dimension [3], [4], [5], [13], as the previous results could not be extended for $d=\mathrm{\Omega }(\sqrt{\log n})$.

The only known lower bounds for rectangle stabbing, when the dimension is non-constant, are based on conditional lower bounds upon the best-known results on CNF-SAT [21]. Therefore, our lower bound is the first non-trivial unconditional lower bound for orthogonal range reporting and rectangle stabbing with non-constant dimension.

我们研究了中等维度的正交范围报告和矩形刺穿问题，即当维度为$C\mathrm{日志}(n)$对于一些常数c。在正交范围报告中，输入是一组d维的n个点，目标是将这n个点存储在数据结构中，这样给定一个查询矩形，我们可以报告矩形中包含的所有输入点。矩形刺穿问题是“对偶”问题，其中输入是一组矩形，查询是一个点。

我们的主要结果如下：假设使用$\mathrm{小号}(n)$空间，我们可以解决任何一个问题$d=C\mathrm{日志}n$方面，$C\ge \mathrm{4个}$， 使用$问(n)+欧(吨)$指针机计算模型中的时间，其中t是输出大小。然后，我们证明如果查询时间小，即$问(n)=n^{\mathrm{1个}-\gamma }$， 为了$\gamma \ge \frac{\mathrm{2个}}{\mathrm{2个}+\mathrm{日志}C}$，那么空间必须是$\mathrm{欧姆}\left(n^{\mathrm{1个}-\gamma }{n}^{\sqrt{C\gamma }/\mathrm{电子}-o(\sqrt{C\gamma })}\right)$. 有趣的是，我们使用非构造性方法获得了这个下界，并且我们展示了一些代码的存在，这些代码概括了纠错码的特定方面。我们的结果克服了先前指针机模型中非恒定维度下限的缺点 [3]、[4]、[5]、[13]，因为先前的结果无法扩展到$d=\mathrm{欧姆}(\sqrt{\mathrm{日志}n})$.

当维度不是常数时，唯一已知的矩形刺穿下界是基于 CNF-SAT [21] 上最著名结果的条件下界。因此，我们的下界是第一个用于正交范围报告和具有非常量维度的矩形刺穿的非平凡无条件下界。