Fast implementation of the Tukey depth

This paper presents two new algorithms for fast calculating the Tukey depth of dimensions $p \ge 3$. The first algorithm is naive, and can compute \emph{exactly} the Tukey depth of a single point with complexity $O\left(n^{p-1}\log(n)\right)$. Compared to the first, the second algorithm further utilizes the \emph{quasiconcave} of the Tukey depth function, and hence can be implemented with complexity less than $O\left(\frac{1}{2^{p-2}} n^{p-1}\log(n)\right)$. Both of them are of \emph{combinatorial} properties. Experiments show that the proposed algorithms require quite minimum memory and run much faster than the existing procedures.