Lower bounds for computing statistical depth
Computational Statistics & Data Analysis
Lower bounds for algebraic computation trees
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Algorithms for bivariate medians and a Fermat--Torricelli problem for lines
Computational Geometry: Theory and Applications - Special issue on the thirteenth canadian conference on computational geometry - CCCG'01
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Let S={s"1,...,s"n} be a set of points in the plane. The Oja depth of a query point @q with respect to S is the sum of the areas of all triangles (@q,s"i,s"j). This depth may be computed in O(nlogn) time in the RAM model of computation. We show that a matching lower bound holds in the algebraic decision tree model. This bound also applies to the computation of the Oja gradient, the Oja sign test, and to the problem of computing the sum of pairwise distances among points on a line.