More Output-Sensitive Geometric Algorithms (Extended Abstract)

Kenneth L. Clarkson

Abstract: A simple idea for speeding up the computation of extrema of a partially ordered set turns out to have a number of interesting applications in geometric algorithms; the resulting algorithms generally replace an appearance of the input size n in the running time by an output size A/spl les/n. In particular, the A coordinate-wise minima of a set of n points in R/sup d/ can be found by an algorithm needing O(nA) time. Given n points uniformly distributed in the unit square, the algorithm needs n+O(n/sup 5/8/) point comparisons on average. Given a set of n points in R/sup d/, another algorithm can find its A extreme points in O(nA) time. Thinning for nearest-neighbor classification can be done in time O(n log n)/spl Sigma//sub i/ A/sub i/n/sub i/, finding the A/sub i/ irredundant points among n/sub i/ points for each class i, where n=/spl Sigma//sub i/ n/sub i/ is the total number of input points. This sharpens a more obvious O(n/sup 3/) algorithm, which is also given here. Another algorithm is given that needs O(n) space to compute the convex hull of n points in O(nA) time. Finally, a new randomized algorithm finds the convex hull of n points in O(n log A) expected time, under the condition that a random subset of the points of size r has expected hull complexity O(r). All but the last of these algorithms has polynomial dependence on the dimension d, except possibly for linear programming.<>