Lower bounds for computing statistical depth

Authors:
Greg Aloupis;Carmen Cortés;Francisco Gómez;Michael Soss;Godfried Toussaint
Affiliations:
School of Computer Science, McGill University, 3480 University Street, Montreal, Que, Canada H3A 2A7;Departamento de Matematica Aplicada, Escuela Universitaria de Ingeniera Tecnica Agricola, Universidad de Sevilla, Spain;Escuela Universitaria de Informatica, Matematica Aplicada, Universidad Politecnica de Madrid, Italy;School of Computer Science, McGill University, 3480 University Street, Montreal, Que, Canada H3A 2A7;School of Computer Science, McGill University, 3480 University Street, Montreal, Que, Canada H3A 2A7
Venue:
Computational Statistics & Data Analysis
Year:
2002

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Abstract

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(nlog 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.