Algorithms for Colourful Simplicial Depth and Medians in the Plane

Abstract

Abstract. The colourful simplicial depth (CSD) of a point x ∈ R 2 relative to a configuration P = (P 1 , P 2 , . . . , P k ) of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point in R 2 , called a median, that maximizes colourful simplicial depth.For computing the colourful simplicial depth of x, our algorithm runs in time O (n log n + kn) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n 4 ). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O (n log n) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n 4 ) for finding a monochrome median.