Output-sensitive results on convex hulls, extreme points, and related problems

We use known data structures for ray-shooting and linear-programming queries to derive new output-sensitive results on convex hulls, extreme points, and related problems. We show that the f-face convex hull of an n-point set P in a fixed dimension d ≥ 2 can be constructed in O(n log f + (nf)^{1−1/(⌊d/2⌋+1)} log^O(1) n) time; this is optimal if f = O(n^1/⌊d/2⌋ / log^K n) for some sufficiently large constant K. We also show that the h extreme points of P can be computed in O(n log^O(1) h + (nh)^{1−1/(⌊d/2⌋+1)} log^O(1) n) time. These results are then applied to produce an algorithm that computes the vertices of all the convex layers of P in O(n^2−γ) time for any constant γ < 2/(⌊d/2⌋² + 1). Finally, we obtain improved time bounds for other problems including levels in arrangements and linear programming with few violated constraints. In all of our algorithms the input is assumed to be in general position.