Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Approximating the Simplicial Depth   Let $P$ be a set of $n$ points in $d$-dimensions. The simplicial depth,
$\sigma_P(q)$ 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 $O(n^{d+1})$ time whereas it
is generally believed that one cannot do better than $O(n^{d-1})$. 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
$(1+\varepsilon)$-approximation of the simplicial depth in time
$\tilde{O}(n^{d/2 + 1})$ ignoring polylogarithmic factor. All of these
algorithms are Monte Carlo algorithms. Furthermore, we present a simple
strategy to compute the simplicial depth exactly in $O(n^d \log n)$ time, which
provides the first improvement over the trivial $O(n^{d+1})$ 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.

37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph