On k-hulls and related problems
SIAM Journal on Computing
Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Topologically sweeping an arrangement
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Computing depth contours of bivariate point clouds
Computational Statistics & Data Analysis - Special issue on classification
An optimal algorithm for hyperplane depth in the plane
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Statistical Models in S
Computing location depth and regression depth in higher dimensions
Statistics and Computing
Hardware-assisted computation of depth contours
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Lower bounds for computing statistical depth
Computational Statistics & Data Analysis
Topological Peeling and Implementation
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Topological Sweep in Degenerate Cases
ALENEX '02 Revised Papers from the 4th International Workshop on Algorithm Engineering and Experiments
An Experimental Study and Comparison of Topological Peeling and Topological Walk
COCOON '02 Proceedings of the 8th Annual International Conference on Computing and Combinatorics
Algorithms for bivariate zonoid depth
Computational Geometry: Theory and Applications
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The concept of location depth was introduced in statistics as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. These require the computation of depth regions, which form a collection of nested polygons. The center of the deepest region is called the Tukey median. The only available implemented algorithms for the depth contours and the Tukey median are slow, which limits their usefulness. In this paper we describe an optimal algorithm which computes all depth contours in &Ogr;(n2) time and space, using topological sweep of the dual arrangement of lines. Once the contours are known, the location depth of any point is computed in &Ogr;(log2 n) time. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.