Cutting hyperplane arrangements

Abstract

We consider a collection H of n hyperplanes in Ed (where the dimension d is fixed). An ε-cutting for H is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all Ed and such that the interior of any simplex is intersected by at most εn hyperplanes of H. We give a deterministic algorithm for finding a (1/r)-cutting with O(rd ) simplices (which is asymptotically optimal). For r≤n1-σ, where δ>0 is arbitrary but fixed, the running time of this algorithm is O(n(log n)O(1) rd-1). In the plane we achieve a time bound O(nr) for r≤n1-δ, 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⊆Ed and a parameter r, we can deterministically compute a (1/r)-net of size O(rlog r) for the range space (X, {X Υ{hooked} R; R is a simplex}), In time O(n(log n)O(1) rd-1 +rO(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ε-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. © 1991 Springer-Verlag New York Inc.