Computing the Planar $\beta$-skeleton Depth

For β ≥ 1, the β-skeleton depth (SkDβ) of a query point q ∈ Rd with respect to a distribution function F on Rd is defined as the probability that q is contained within the β-skeleton influence region of a random pair of points from F . The β-skeleton depth of q ∈ Rd can also be defined with respect to a given data set S ⊆ Rd. In this case, computing the β-skeleton depth is based on counting all of the β-skeleton influence regions, obtained from pairs of points in S, that contain q. The β-skeleton depth introduces a family of depth functions that contains spherical depth and lens depth for β = 1 and β = 2, respectively. The straightforward algorithm for computing the β-skeleton depth in dimension d takes O(dn2). This complexity of computation is a significant advantage of using the β-skeleton depth in multivariate data analysis because unlike most other data depths, the time complexity of the β-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 Θ(n log n) for computing the planar spherical depth, and the second algorithm with the time complexity of O(n 3 2 + ) is for computing the planar β-skeleton depth, β > 1. By reducing the problem of Element Uniqueness, we prove that computing the β-skeleton depth requires Ω(n log n) time. Some geometric properties of β-skeleton depth are also investigated in this paper. These properties indicate that simplicial depth (SD) is linearly bounded by β-skeleton depth (in particular, SkDβ ≥ 23SD; β ≥ 1). To illustrate this relationship, the results of some experiments on random point sets are provided. In these experiments, the bounds of SphD ≥ 2SD and LD ≥ 1.2SphD are achieved.