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

Authors:
Greg Aloupis;Stefan Langerman;Michael Soss;Godfried Toussaint
Affiliations:
School of Computer Science, McGill University, 3480 University Street, rm. 318, Montreal, QC, H3A 2A7, Canada;School of Computer Science, McGill University, 3480 University Street, rm. 318, Montreal, QC, H3A 2A7, Canada;Chemical Computing Group, 1010 Sherbrooke St, Montreal, QC, H3A 2R7, Canada;School of Computer Science, McGill University, 3480 University Street, rm. 318, Montreal, QC, H3A 2A7, Canada
Venue:
Computational Geometry: Theory and Applications - Special issue on the thirteenth canadian conference on computational geometry - CCCG'01
Year:
2003

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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(n log3 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(n4) time. This median is a point in R2 which is contained in the most triangles formed by elements of S.