Algorithms for Colourful Simplicial Depth and Medians in the Plane

The colourful simplicial depth (CSD) of a point x ∈ R relative to a configuration P = (P , P , . . . , P ) 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, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O (n logn+ 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). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O (n logn) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n) for finding a monochrome median.