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

2026-05-09

AI Summary

This work resolves a long-standing complexity challenge concerning the vertical decomposition of arrangements of semi-algebraic sets in three- and four-dimensional spaces. By integrating techniques from computational geometry—specifically vertical decomposition, arrangement theory, and semi-algebraic set analysis—it establishes, for the first time, nearly tight upper bounds on the complexity of the vertical decomposition of three-dimensional complements and the minimization diagrams of trivariate functions. Building upon this theoretical breakthrough, the study introduces efficient algorithms for constructing vertical decompositions and (1/r)-cuttings, and develops novel data structures that support point-location queries in both three and four dimensions, thereby resolving several longstanding open problems in the field.

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 ${\mathbb R}^4$.

Problem

• vertical decomposition
• semi-algebraic sets
• arrangements
• complexity bounds
• minimization diagram

Innovation

• vertical decomposition
• semi-algebraic sets
• arrangements
• complexity bounds
• point-enclosure queries