Nearly-Tight Bounds for Vertical Decomposition in Three and Four Dimensions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915998199709696 |
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| author | Agarwal, Pankaj K. Ezra, Esther Sharir, Micha |
| author_facet | Agarwal, Pankaj K. Ezra, Esther Sharir, Micha |
| contents | 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 ${\mathbb R}^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09234 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nearly-Tight Bounds for Vertical Decomposition in Three and Four Dimensions Agarwal, Pankaj K. Ezra, Esther Sharir, Micha Computational Geometry 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 ${\mathbb R}^4$. |
| title | Nearly-Tight Bounds for Vertical Decomposition in Three and Four Dimensions |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2605.09234 |