Vertical Decomposition in 3D and 4D with Applications to Line Nearest-Neighbor Searching in 3D

Abstract

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 $\mathcal{U}({\mathcal{S}}^{\prime })$ of any subset ${\mathcal{S}}^{\prime }\subseteq \mathcal{S}$ of size at most $m$. We prove that the complexity of the vertical decomposition of the complement of $\mathcal{U}(\mathcal{S})$ is $O^{\ast }(n^2+U(n))$ (where the $O^{\ast }(\cdot )$ notation hides subpolynomial factors). We also show that the complexity of the vertical decomposition of the entire arrangement $\mathcal{A}(\mathcal{S})$ is $O^{\ast }(n^2+X)$, 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 $O^{\ast }(n^3)$. 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.