Recently, Ezra and Sharir [ES22a] showed an $O(n^{3/2+\sigma})$ space and $O(n^{1/2+\sigma})$ query time data structure for ray shooting among triangles in $\mathbb{R}^3$. This improves the upper bound given by the classical $S(n)Q(n)^4=O(n^{4+\sigma})$ space-time tradeoff for the first time in almost 25 years and in fact lies on the tradeoff curve of $S(n)Q(n)^3=O(n^{3+\sigma})$. However, it seems difficult to apply their techniques beyond this specific space and time combination. This pheonomenon appears persistently in almost all recent advances of flat object intersection searching, e.g., line-tetrahedron intersection in $\mathbb{R}^4$ [ES22b], triangle-triangle intersection in $\mathbb{R}^4$ [ES22b], or even among flat semialgebraic objects [AAEKS22]. We give a timely explanation to this phenomenon from a lower bound perspective. We prove that given a set $\mathcal{S}$ of $(d-1)$-dimensional simplicies in $\mathbb{R}^d$, any data structure that can report all intersections with small ($n^{o(1)}$) query time must use $\Omega(n^{2(d-1)-o(1)})$ space. This dashes the hope of any significant improvement to the tradeoff curves for small query time and almost matches the classical upper bound. We also obtain an almost matching space lower bound of $\Omega(n^{6-o(1)})$ for triangle-triangle intersection reporting in $\mathbb{R}^4$ when the query time is small. Along the way, we further develop the previous lower bound techniques by Afshani and Cheng [AC21, AC22].

翻译：最近，Ezra 和 Sharir [ES22a] 提出了一种用于 $\mathbb{R}^3$ 中三角形间射线射击的数据结构，其空间复杂度为 $O(n^{3/2+\sigma})$，查询时间复杂度为 $O(n^{1/2+\sigma})$。这是近25年来首次对经典空间-时间权衡 $S(n)Q(n)^4=O(n^{4+\sigma})$ 的上界进行改进，且实际上该结果落在 $S(n)Q(n)^3=O(n^{3+\sigma})$ 的权衡曲线上。然而，将其技术推广至其他特定空间与时间组合似乎存在困难。这一现象持续出现在几乎所有近期关于平面对象相交搜索的进展中，例如 $\mathbb{R}^4$ 中的线-四面体相交 [ES22b]、$\mathbb{R}^4$ 中的三角形-三角形相交 [ES22b]，甚至平面半代数对象间的相交问题 [AAEKS22]。我们从下界角度对这一现象给出了及时的解释。我们证明：给定 $\mathbb{R}^d$ 中一组 $(d-1)$ 维单纯形 $\mathcal{S}$，任何能够以较小（$n^{o(1)}$）查询时间报告所有相交的数据结构必须使用 $\Omega(n^{2(d-1)-o(1)})$ 的空间。这破灭了在较小查询时间下显著改进权衡曲线的希望，且该下界几乎匹配经典上界。此外，对于 $\mathbb{R}^4$ 中的三角形-三角形相交报告问题，当查询时间较小时，我们得到几乎匹配的空间下界 $\Omega(n^{6-o(1)})$。在此过程中，我们进一步推广了 Afshani 和 Cheng [AC21, AC22] 先前提出的下界技术。