Point set stratification and Delaunay depth

TL;DR

This paper analyzes the Delaunay depth function, comparing its sensitivity to multimodality with other depth functions, and develops an efficient algorithm for computing Delaunay depth contours in planar point sets.

Contribution

It introduces an analysis of Delaunay depth stratification, compares it with other depths, and presents an optimal algorithm for computing depth contours in the plane.

Findings

Delaunay depth is sensitive to multimodality.

An O(n log^2 n) algorithm for depth contours is developed.

Computing Delaunay depth in O(n log n) is proven to be optimal.

Abstract

In the study of depth functions it is important to decide whether we want such a function to be sensitive to multimodality or not. In this paper we analyze the Delaunay depth function, which is sensitive to multimodality and compare this depth with others, as convex depth and location depth. We study the stratification that Delaunay depth induces in the point set (layers) and in the whole plane (levels), and we develop an algorithm for computing the Delaunay depth contours, associated to a point set in the plane, with running time O(n log^2 n). The depth of a query point p with respect to a data set S in the plane is the depth of p in the union of S and p. When S and p are given in the input the Delaunay depth can be computed in O(n log n), and we prove that this value is optimal.