Approximating the $k$-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 O(1/r)cutting of the first n/r levels of A(H), which consists of O(r) semi-unbounded vertical triangular prisms. The same construction yields an approximation of the (n/r)level by a terrain consisting of O(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 O(n + r2 ε−6 log3 r) expected time. An application of this technique allows us to mimic Matouˇsek’s construction of cuttings in the plane [36], to obtain a similar construction of “layered” (1/r)-cutting of the entire arrangement A(H), of optimal size O(r3 ). Another ap...

• Type: Journal
• Year: 2016
• Where: CORR
• Authors: Sariel Har-Peled, Haim Kaplan, Micha Sharir
• Keywords: CORR 2016, Education