Approximating the Simplicial Depth

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Let P be a set of n points in d-dimensions. The simplicial depth, σₚ(q) of a point q is the number of d-simplices with vertices in P that contain q in their convex hulls. The simplicial depth is a notion of data depth with many applications in robust statistics and computational geometry. Computing the simplicial depth of a point is known to be a challenging problem. The trivial solution requires O(nᵈ⁺¹) time whereas it is generally believed that one cannot do better than O(nᵈ⁻¹). In this paper, we consider approximation algorithms for computing the simplicial depth of a point. For d=2, we present a new data structure that can approximate the simplicial depth in polylogarithmic time, using polylogarithmic query time. In 3D, we can approximate the simplicial depth of a given point in near-linear time, which is clearly optimal up to polylogarithmic factors. For higher dimensions, we consider two approximation algorithms with different worst-case scenarios. By combining these approaches, we compute a (1+ε)-approximation of the simplicial depth in time O(nᵈ/² ⁺ ¹) ignoring polylogarithmic factor. All of these algorithms are Monte Carlo algorithms. Furthermore, we present a simple strategy to compute the simplicial depth exactly in O(nᵈ n) time, which provides the first improvement over the trivial O(nᵈ⁺¹) time algorithm for d>4. Finally, we show that computing the simplicial depth exactly is #P-complete and W[1]-hard if the dimension is part of the input.