An optimal-time algorithm for slope selection
SIAM Journal on Computing
Computing a centerpoint of a finite planar set of points in linear time
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
The discrepancy method: randomness and complexity
The discrepancy method: randomness and complexity
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
An optimal randomized algorithm for maximum Tukey depth
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
An optimal extension of the centerpoint theorem
Computational Geometry: Theory and Applications
Centerpoints and Tverberg's technique
Computational Geometry: Theory and Applications
Hitting Simplices with Points in ℝ3
Discrete & Computational Geometry
Overlap properties of geometric expanders: extended abstract
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Over the past several decades, many combinatorial measures have been devised for capturing the statistical data depth of a set of n points in R2. These include Tukey depth [15], Oja depth [12], Simplicial depth [10] and several others. Recently Fox et al. [7] have defined the Ray-Shooting depth of a point set, and given a topological proof for the existence of points with high Ray-Shooting depth in R2. In this paper, we present an O(n2log2n)-time algorithm for computing a point of high Ray-Shooting depth. We also present a linear time approximation algorithm.