Weighted orthogonal linear L ∞-approximation and applications

Abstract

Let S={s1, s2,…,sn} be a set of sites in Ed, where every site si has a positive real weight ωi. This paper gives an algorithm to find a weighted orthogonal L∞-approximation hyperplane for S. The algorithm is shown to require O(nlogn) time and O(n) space for d=2, and O(n[d/2+1]) time and O(n[(d+1)/2]) space for d>2. The L∞-approximation algorithm will be adapted to solve the problem of finding the width of a set of n points in Ed, and the problem of finding a stabbing hyperplane for a set of n hyperspheres in Ed with varying radii. The time and space complexities of the width and stabbing algorithms are seen to be the same as those of the L∞-approximation algorithm.