Approximating the a;-level in three-dimensional plane arrangements

Let H be a set of n non-vertical planes in three dimensions, and let r < n be a parameter. We give a simple alternative proof of the existence of a 0(1/r)-cutting of the first n/r levels of A(H), which consists of 0(r) semi-unbounded vertical triangular prisms. The same construction yields an approximation of the (n/r)-level by a terrain consisting of 0(r/ϵ3) triangular faces, which lies entirely between the levels (1 ± ϵ)n/r. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. The proof is constructive, and leads to a simple randomized algorithm, that computes the terrain in 0(n + r2ϵ-6 log3 r) expected time. An application of this technique allows us to mimic Matousek's construction of cuttings in the plane [36], to obtain a similar construction of "layered" (l/r)-cutting of the entire arrangement A(H), of optimal size 0(r3). Another application is a simplified optimal approximate range counting algorithm in three dimensions, competing with that of Afshani and Chan [1].