Given a finite set of points S, two measures of the depth of a query point θ with respect to S are the Simplicial depth of Liu and the Halfspace depth of Tukey (also known as Location depth). We show that computing these depths requires Ω(n log n) time, which matches the upper bound complexities of the algorithms of Rousseeuw and Ruts. Our lower bound proofs may also be applied to two bivariate sign tests: that of Hodges, and that of Oja and Nyblom.
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