Nearly-Tight Bounds for Vertical Decomposition in Three and Four Dimensions

TL;DR

This paper establishes nearly-tight bounds for vertical decompositions in three and four dimensions, resolving longstanding open problems and enabling efficient algorithms and data structures for geometric problems.

Contribution

It provides sharp bounds on the complexity of vertical decompositions in 3D and 4D, leading to improved algorithms and data structures for related geometric problems.

Findings

Sharp bounds on the complexity of vertical decompositions in 3D and 4D.

Efficient algorithms for constructing decompositions and cuttings.

Data structures for point-enclosure queries in 3D and 4D.

Abstract

Vertical decomposition is a widely used general technique for decomposing the cells of arrangements of semi-algebraic sets in ${{\mathbb R}}^d$ 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 $d = 3, 4$. For example, we obtain sharp bounds on the complexity of the vertical decomposition of the complement of the union of a set of semi-algebraic regions of constant complexity in ${{\mathbb R}}^3$, and of the minimization diagram of a set of trivariate functions. These results lead to efficient algorithms for a variety of problems involving vertical decompositions, including algorithms for constructing the decompositions themselves and for constructing $(1/r)$-cuttings of substructures of arrangements. They also lead to a data structure for answering point-enclosure queries amid semi-algebraic sets in ${{\mathbb R}}^3$ and ${{\mathbb R}}^4$.