Computing the Planar $\beta$-skeleton Depth

For $\beta \geq 1$, the \emph{$\beta$-skeleton depth} ($\SkD_\beta$) of a query point $q\in \mathbb{R}^d$ with respect to a distribution function $F$ on $\mathbb{R}^d$ is defined as the probability that $q$ is contained within the \emph{$\beta$-skeleton influence region} of a random pair of points from $F$. The $\beta$-skeleton depth of $q\in \mathbb{R}^d$ can also be defined with respect to a given data set $S\subseteq \mathbb{R}^d$. In this case, computing the $\beta$-skeleton depth is based on counting all of the $\beta$-skeleton influence regions, obtained from pairs of points in $S$, that contain $q$. The $\beta$-skeleton depth introduces a family of depth functions that contains \emph{spherical depth} and \emph{lens depth} for $\beta=1$ and $\beta=2$, respectively. The straightforward algorithm for computing the $\beta$-skeleton depth in dimension $d$ takes $O(dn^2)$. This complexity of computation is a significant advantage of using the $\beta$-skeleton depth in multivariate data analysis because unlike most other data depths, the time complexity of the $\beta$-skeleton depth grows linearly rather than exponentially in the dimension $d$. The main results of this paper include two algorithms. The first one is an optimal algorithm that takes $\Theta(n\log n)$ for computing the planar spherical depth, and the second algorithm with the time complexity of $O(n^{\frac{3}{2}+\epsilon})$ is for computing the planar $\beta$-skeleton depth, $\beta >1$. By reducing the problem of \textit{Element Uniqueness}, we prove that computing the $\beta$-skeleton depth requires $\Omega(n \log n)$ time. Some geometric properties of $\beta$-skeleton depth are also investigated in this paper. These properties indicate that \emph{simplicial depth} ($\SD$) is linearly bounded by $\beta$-skeleton depth. Some experimental bounds for different depth functions are also obtained in this paper.