## 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.

Original language | English (US) |
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Pages (from-to) | 69-79 |

Number of pages | 11 |

Journal | Computational Geometry: Theory and Applications |

Volume | 26 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2003 |

## Keywords

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

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics