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

TL;DR

This paper establishes a tight (n log n) lower bound for computing the sum of even-ranked elements in a sequence, resolving an open problem in computational geometry.

Contribution

It proves the (n log n) lower bound for the Even-Rank-Sum problem in the algebraic computation-tree model, confirming the optimal complexity.

Findings

The problem requires (n log n) time in the algebraic model.

The result resolves a longstanding open problem from 2008.

It demonstrates the optimality of existing algorithms for this problem.

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 \Theta(n log n). This solves an open problem posed by Michael Shamos at the Canadian Conference on Computational Geometry in 2008.