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.

Point set stratification and Delaunay depth

“…In particular, we see that the depth of a point p with respect to a set of data S = {s 1 , · · · , s n } can be found in O(n log n) time. Lower bounds for this kind of problems have attracted significant attention, and in Section 4 we carry out a study similar to those by Aloupis et al in [ACG + 02] and [AMcL04], proving an Ω(n log n) lower bound for Delaunay depth computation.…”

“…When S and p are the entry data, the Tukey depth of p, its simplicial depth and its Oja depth can be computed in O(n log n) [RR96]. In [ACG + 02] it was proved that this value is also a tight bound for the first two cases and recently it has been proved an identical result for the Oja depth [AMcL04] . The convex depth of p can be easily computed in O(n log n) time, since it suffices to find the layers of S ∪ {p}, and it is easy to see that this value is tight.…”

“…When S and p are the entry data, the Tukey depth of p, its simplicial depth and its Oja depth can be computed in O(n log n) [RR96]. In [ACG + 02] it was proved that this value is also a tight bound for the first two cases and recently it has been proved an identical result for the Oja depth [AMcL04] .…”

Geometric measures of data depth

“…With the incorporation of a powerful optimization technique for arrangements of lines, developed by Langerman and Steiger [LS03], the time complexity of computing the Oja median was improved to O(n log 3 n) time [ALST03]. An Ω(n log n) lower bound for computing the Oja depth of a point is to appear soon [AM04].…”

Algorithms for bivariate zonoid depth

“…The design of efficient algorithms for these problems is essential for these depth measures to become useful statistical analysis tools. Computational geometry [32] has been of great help in this respect, and there are many results in the computational geometry literature [1][2][3][4][5][9][10][11][12]14,17,19,[21][22][23]26,29,31,33,34,36,37].…”