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 R 3 , and let U(m) be an upper bound on the complexity of the union U (S ′ ) of any subset S ′ ⊆ 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. Finally, We study batched, or offline, variants of these problems, and obtain improved algorithms for such scenarios.