## Abstract

Vertical decomposition is a widely used general technique for decomposing the cells of arrangements of semi-algebraic sets in 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: (i) Let S be a collection of n semi-algebraic sets of constant complexity in R^{3}, and let U(m) be an upper bound on the complexity of the union U(S^{0}) of any subset S^{0} ⊆ S of size at most m. We prove that the complexity of the vertical decomposition of the complement of U(S) is O^{∗}(n^{2} + U(n)) (where the O^{∗}(·) notation hides subpolynomial factors). We also show that the complexity of the vertical decomposition of the entire arrangement A(S) is O^{∗}(n^{2} + X), where X is the number of vertices in A(S). (ii) Let 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 A(F) in R^{4} lying below the lower envelope of F is O^{∗}(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 R^{3}: We first present a linear-size data structure for answering nearest-neighbor queries, with points, amid n lines in R^{3} in O^{∗}(n^{2}/^{3}) time per query. We also study the converse problem, where we return the nearest neighbor of a query line amid n input points, or lines, in R^{3}. We obtain a data structure of O^{∗}(n^{4}) size that answers a nearest-neighbor query in O(log n) time.

Original language | English |
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Pages | 150-170 |

Number of pages | 21 |

DOIs | |

State | Published - 2024 |

Event | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States Duration: 7 Jan 2024 → 10 Jan 2024 |

### Conference

Conference | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 |
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Country/Territory | United States |

City | Alexandria |

Period | 7/01/24 → 10/01/24 |