Tight Bounds on the Complexity of Recognizing Odd-Ranked Elements

TL;DR

This paper establishes tight bounds of Theta(n log n) on the computational complexity for recognizing whether odd-positioned elements in a vector are exactly those with odd ranks, within the algebraic computation tree model.

Contribution

It provides the first tight bounds on the complexity of the recognition problem for odd-ranked elements in a vector.

Findings

Proves a Theta(n log n) bound on the problem's complexity.

Establishes the problem's computational difficulty in the algebraic model.

Abstract

Let S = <s_1, s_2, s_3, ..., s_n> be a given vector of n real numbers. The rank of a real z with respect to S is defined as the number of elements s_i in S such that s_i is less than or equal to z. We consider the following decision problem: determine whether the odd-numbered elements s_1, s_3, s_5, ... are precisely the elements of S whose rank with respect to S is odd. We prove a bound of Theta(n log n) on the number of operations required to solve this problem in the algebraic computation tree model.