Construction of ɛ-nets
LetS be a set ofn points in the plane and letɛ be a real number, 0<ɛ<1. We give a deterministic algorithm, which in timeO(nɛ−2 log(1/ɛ)+ɛ−8) (resp.O(nɛ−2 log(1/ɛ)+ɛ−10) constructs anɛ-netN⊂S of sizeO((1/ɛ) (log(1/ɛ))2) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatɛn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).
We also prove that given a setL ofn lines in the plane, we can cut the plane intoO(ɛ−2) triangles in such a way that no triangle is intersected by more thanɛn lines ofL. We give a deterministic algorithm for this with running timeO(nɛ−2 log(1/ɛ)). This has numerous applications in various computational geometry problems.