A lower bound for computing Oja depth

Abstract

Let S = { s 1 , … , s n } be a set of points in the plane. The Oja depth of a query point θ with respect to S is the sum of the areas of all triangles ( θ , s i , s j ) . This depth may be computed in O ( n log n ) 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.