Lower bounds for finding the maximum and minimum elements with k lies

TL;DR

This paper establishes lower bounds on the number of comparisons needed to find the maximum and minimum elements in a totally ordered set when up to k comparisons may be erroneous, challenging previous conjectures.

Contribution

It proves that at least (k+1.5)n + Θ(k) comparisons are necessary, providing a new lower bound that refutes earlier conjectures about sufficiency.

Findings

At least (k+1.5)n + Θ(k) comparisons are required in the worst case.

The paper disproves the conjecture that (k+1+ε)n comparisons suffice.

It advances understanding of comparison-based algorithms under error constraints.

Abstract

In this paper we deal with the problem of finding the smallest and the largest elements of a totally ordered set of size $n$ using pairwise comparisons if $k$ of the comparisons might be erroneous where $k$ is a fixed constant. We prove that at least $(k+1.5)n+\Theta(k)$ comparisons are needed in the worst case thus disproving the conjecture that $(k+1+\epsilon)n$ comparisons are enough.