An Ω(n log n) lower bound for computing the sum of even-ranked elements

Abstract

Given a sequence A of 2n real numbers, the Even-Rank-Sum problem asks for the sum of the n values that are at the even positions in the sorted order of the elements in A. We prove that, in the algebraic computation-tree model, this problem has time complexity Θ(n log n). This solves an open problem posed by Michael Shamos at the Canadian Conference on Computational Geometry in 2008.