Algorithms for Colourful Simplicial Depth and Medians in the Plane

Abstract

The colourful simplicial depth (CSD) of a point relative to a configuration $P=(P^1, P^2, \ldots , P^k)$ 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 , called a median, that maximizes colourful simplicial depth.

For computing the colourful simplicial depth of $x$, our algorithm runs in time $O\left( n \log {n} + k n \right)$ 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 $O\left( n \log {n} \right)$ 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.