Algorithms for bivariate medians and a Fermat–Torricelli problem for lines

A lower bound for computing Oja depth

“…Such notions are of great interest to the statistical community. We refer the reader to [1,2] for recent results with a similar flavor to what is presented here, and for further introductory references to the topic of statistical depth. The main result in [1] involves lower bounds on the computation of halfspace depth [8] and simplicial depth [5].…”

“…This gradient may be computed in O(n log n) time [7]. It is used for the computation of the Oja median, which is the point in the plane that has 5 highest Oja depth (see [2]). Suppose that we are given a planar point set that is contained on a line .…”

The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg

“…Every set of n points in R d , d ≥ 3, has Oja depth at most Algorithms. For the case d = 2, Rousseeuw and Ruts [329] presented a O(n 5 log n) time algorithm for computing the lowest depth point, which was then improved to the current-best algorithm with running time O(n log 3 n) [25]. A point of Oja depth at most n 2 9 can be computed in O(n log n) time [297].…”

Multivariate Medians for Image and Shape Analysis

“…Building on the geometric property mentioned above, [45,53] develop algorithms that allow to compute the exact Oja median of N data points in n dimensions with O(nN n log N ), or an approximation with stochastic accuracy guarantee in O(5 n /ε 2 ) where ε is a confidence radius in L ∞ sense. An O(N log 3 N ) algorithm for the bivariate Oja median is stated in [1,2].…”