Approximating the k-Level in Three-Dimensional Plane Arrangements [article]

Sariel Har-Peled, Haim Kaplan, Micha Sharir
2016 arXiv   pre-print
]#1(#2) Let H be a set of n planes in three dimensions, and let r ≤ n be a parameter. We give a simple alternative proof of the existence of a (1/r)-cutting of the first n/r levels of (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/^3) triangular faces, which lies entirely between the levels (1±)n/r. The proof does not use sampling, and exploits techniques based on planar
more » ... tors 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, with expected near-linear running time. An application of this technique allows us to mimic Matousek's construction of cuttings in the plane, to obtain a similar construction of "layered" (1/r)-cutting of the entire arrangement (H), of optimal size O(r^3). Another application is a simplified optimal approximate range counting algorithm in three dimensions, competing with that of Afshani and Chan.
arXiv:1601.04755v2 fatcat:ueg4qasfbrbi5n4vh5ruztdtma