Simplicial depth measures how deep a point is among a set of points. Efficient algorithms to
compute it are important to its usefulness in applications, such as multivariate analysis in
environmental health and bioinformatics. When the points are in Ed, a straightforward
method takes O (nd+ 1) time. We discuss an algorithm that takes O (n 2) time when the
points are in E 3, and we generalize it to E 4 with a time complexity of O (n 4).

Simplicial depth is a way to measure how deep a point is among a set of points. E cient
algorithms to compute it are important to the usefulness of its applications, such as in
multivariate analysis in statistics. A straightforward method takes O (nd+ 1) time when the
points are in d-dimensional space. We discuss an algorithm that takes O (n2) time when the
points are in three-dimensional space, and we generalize it to four-dimensional space with a
time complexity of O (n4). For spaces higher than four-dimensional, there are no known …