CUTTING HYPERPLANE ARRANGEMENTS

We consider a collection H of n hyperplanes in E(d) (where the dimension d is fixed). An epsilon-cutting for H is a collection of (possibly unbounded) d-dimensional simplices with disjoint interiors, which cover all E(d) and such that the interior of any simplex is intersected by at most epsilon-n hyperplanes of H. We give a deterministic algorithm for finding a (1/r)-cutting with O(r(d)) simplices (which is asymptotically optimal). For r less-than-or-equal-to n 1-delta, where delta > 0 is arbitrary but fixed, the running time of this alogorithm is O(n(log n)O(1)r(d-1)). In the plane we achieve a time bound O(nr) for r less-than-or-equal-to n 1-delta, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For an n point set X is-contained-in-or-equal-to E(d) and a parameter r, we can deterministically compute a (1/r)-net of size O(r log r) for the range space (X, {X intersection R; R is a simplex}), in time O(n(log n)O(1)r(d-1) + r(O(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find epsilon-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly.