Nearly-Tight Bounds for Vertical Decomposition in Three and Four Dimensions
Provides sharp theoretical bounds for a fundamental technique in computational geometry, enabling improved algorithms for arrangements of semi-algebraic sets.
The paper resolves long-standing open problems by proving nearly-tight bounds on the complexity of vertical decomposition for substructures of arrangements in 3D and 4D, leading to efficient algorithms and a data structure for point-enclosure queries.
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 ${\mathbb R}^4$.