We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n ) time and space algorithm computes directed depth of all points in d dimensions. Properties of undirected depth lead to an O(n log n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(n log n) time algorithm by Langerman and Steiger . Furthermore, we describe the structure of depth in the plane and higher dimensions and also give approximation algorithms for hyperplane arrangements and degenerate line arrangements
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