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

Abstract

Given a set S of n points in R 2 , the Oja depth of a point θ is the sum of the areas of all triangles formed by θ 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( nlog 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.