A data depth can be used to measure the “depth” or “outlyingness” of a given multivariate
sample with respect to its underlying distribution. This leads to a natural center-outward …

We discuss five fundamental results of discrete mathematics: the lemmas of Sperner and
Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg …

We present the first optimal algorithm to compute the maximum Tukey depth (also known as
location or halfspace depth) for a non-degenerate point set in the plane. The algorithm is …

Several measures of data depth have been proposed, each attempting to maintain certain
robustness properties. This paper lists the main approaches known to the computer science …

This book surveys the mathematical and computational properties of finite sets of points in
the plane, covering recent breakthroughs on important problems in discrete geometry, and …

Given a set S of n points in R 2, the Oja depth of a point θ is the sum of the areas of all
triangles formed by θ and two elements of S. A point in R 2 with minimum depth is an Oja …

We present the Iterated-Tverberg algorithm, the first deterministic algorithm for computing an
approximate centerpoint of a set S∈ Rd with running time sub-exponential in d. The …

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 …

We define quasiconvex programming, a form of generalized linear programming in which
one seeks the point minimizing the pointwise maximum of a collection of quasiconvex …

We evaluate the practical efficiency of a new shortest path algorithm for undirected graphs
which was developed by the first two authors. This algorithm works on the fundamental …