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(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* (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*(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 ℝ³. We obtain a data structure of O*(n⁴) size that answers a nearest-neighbor query in O(log n) time.

^* Work by Pankaj Agarwal has been partially supported by NSF grants IIS-18-14493, CCF-20-07556, and CCF-22-23870. Work by Esther Ezra has been partially supported by Israel Science Foundation Grant 800/22, and also by US-Israel Binational Science Foundation under Grant 2022131. Work by Micha Sharir has been partially supported by Israel Science Foundation Grant 260/18.