Lower Bounds for Intersection Reporting among Flat Objects

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].