Vertical decomposition is a widely used general technique for decomposing the cells of arrangements of semi-algebraic sets in $d$-space into constant-complexity subcells. In this paper, we settle in the affirmative a few long-standing open problems involving the vertical decomposition of substructures of arrangements for $d=3,4$: (i) Let $\mathcal{S}$ be a collection of $n$ semi-algebraic sets of constant complexity in 3D, and let $U(m)$ be an upper bound on the complexity of the union $\mathcal{U}(\mathcal{S}')$ of any subset $\mathcal{S}'\subseteq \mathcal{S}$ of size at most $m$. We prove that the complexity of the vertical decomposition of the complement of $\mathcal{U}(\mathcal{S})$ is $O^*(n^2+U(n))$ (where the $O^*(\cdot)$ notation hides subpolynomial factors). We also show that the complexity of the vertical decomposition of the entire arrangement $\mathcal{A}(\mathcal{S})$ is $O^*(n^2+X)$, where $X$ is the number of vertices in $\mathcal{A}(\mathcal{S})$. (ii) Let $\mathcal{F}$ be a collection of $n$ trivariate functions whose graphs are semi-algebraic sets of constant complexity. We show that the complexity of the vertical decomposition of the portion of the arrangement $\mathcal{A}(\mathcal{F})$ in 4D lying below the lower envelope of $\mathcal{F}$ is $O^*(n^3)$. These results lead to efficient algorithms for a variety of problems involving these decompositions, including algorithms for constructing the decompositions themselves, and for constructing $(1/r)$-cuttings of substructures of arrangements of the kinds considered above. One additional algorithm of interest is for output-sensitive point enclosure queries amid semi-algebraic sets in three or four dimensions. In addition, as a main domain of applications, we study various proximity problems involving points and lines in 3D.

翻译：垂直分解是一种广泛使用的通用技术，用于将$d$维空间中半代数集排列的单元格分解为常数复杂度的子单元格。本文肯定地解决了几个长期悬而未决的开放问题，涉及$d=3,4$时排列子结构的垂直分解：（i）设$\mathcal{S}$为3D中$n$个常数复杂度半代数集的集合，$U(m)$为任意大小不超过$m$的子集$\mathcal{S}'\subseteq \mathcal{S}$的并集$\mathcal{U}(\mathcal{S}')$复杂度的上界。我们证明$\mathcal{U}(\mathcal{S})$补集的垂直分解复杂度为$O^*(n^2+U(n))$（其中$O^*(\cdot)$表示次多项式因子被隐藏）。我们还证明整个排列$\mathcal{A}(\mathcal{S})$的垂直分解复杂度为$O^*(n^2+X)$，其中$X$为$\mathcal{A}(\mathcal{S})$中的顶点数。（ii）设$\mathcal{F}$为$n个$三元函数的集合，其图形为常数复杂度的半代数集。我们证明4D中排列$\mathcal{A}(\mathcal{F})$位于$\mathcal{F}$下包络以下部分的垂直分解复杂度为$O^*(n^3)$。这些结果催生了针对这些分解的多种问题的高效算法，包括构建分解本身的算法，以及构建上述排列子结构$(1/r)$-切割的算法。另一个值得关注的算法是三维或四维半代数集中输出敏感的包围点查询算法。此外，作为主要应用领域，我们研究了3D中涉及点和线的各种邻近问题。