Vertical Decomposition in 3D and 4D with Applications to Line Nearest-Neighbor Searching in 3D

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 R³, and let U(m) be an upper bound on the complexity of the union U(S⁰) of any subset S⁰ ⊆ S of size at most m. We prove that the complexity of the vertical decomposition of the complement of U(S) is O^∗(n² + 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^∗(n² + 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 R⁴ lying below the lower envelope of F is O^∗(n³). 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 R³: We first present a linear-size data structure for answering nearest-neighbor queries, with points, amid n lines in R³ in O^∗(n²/³) 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 R³. We obtain a data structure of O^∗(n⁴) size that answers a nearest-neighbor query in O(log n) time.