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

https://doi.org/10.1016/S0925-7721(02)00173-6Get rights and content
Under an Elsevier user license
open archive

Abstract

Given a set S of n points in R2, 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 R2 with minimum depth is an Oja median. We show how an Oja median may be computed in O(nlog3n) 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(n4) time. This median is a point in R2 which is contained in the most triangles formed by elements of S.

Keywords

Oja depth
Simplicial depth
Fermat–Torricelli problem
Geometric medians
Estimators of location
Algorithms
Computational geometry

Cited by (0)