ORTHOGONAL WEIGHTED LINEAR L1 AND L-INFINITY APPROXIMATION AND APPLICATIONS

Let S = { s1,s2,..., s(n)} be a set of sites in E(d), where every site s(i) has a positive real weight omega(i). This paper gives algorithms to find weighted orthogonal L(infinity) and L1 approximating hyperplanes for S. The algorithm for the weighted orthogonal L1 approximation is shown to require O(n(d)) worst-case time and O(n) space for d greater-than-or-equal-to 2. The algorithm for the weighted orthogonal L(infinity) approximation is shown to require 0(n log n) worst-case time and 0(n) space for d = 2, and O(n(right perpendicular d/2 + 1 left perpendicular)) worst-case time and 0(n(right perpendicular (d + 1)/2 left perpendicular)) space for d > 2. In the latter case, the expected time complexity may be reduced to O(n(right perpendicular (d + 1)/2 left perpendicular)). The L(infinity) approximation algorithm can be modified to solve the problem of finding the width of a set of n points in E(d), and the problem of finding a stabbing hyperplane for a set of n hyperspheres in E(d) 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(infinity) approximation algorithm.