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

Michiel Smid

Given a sequence A of 2n real numbers, the EvenRankSum 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(nlog n). This solves an open problem posed by Michael Shamos at the Canadian Conference on Computational Geometry in 2008. (Joint work with Marc Mörig, Dieter Rautenbach, and Jan Tusch.)