Algorithms for bivariate medians and a Fermat-Torricelli problem for lines

Given a set S of n points in R-2, the Oja depth of a point theta is the sum of the areas of all triangles formed by theta and two elements of S. A point in R-2 with minimum depth is an Oja median. We show how an Oja median may be computed in O(n log(3) n) time. In addition, we present an algorithm for computing the Fermat-Torricelli points of n lines in O(n) time. These points minimize the sum of weighted distances to the lines. Finally, we propose an algorithm which computes the simplicial median of S in O(n(4)) time. This median is a point in R-2 which is contained in the most triangles formed by elements of S. (C) 2003 Elsevier Science B.V. All rights reserved.