Minimum and maximum against k lies

TL;DR

This paper investigates the problem of finding minimum and maximum elements in a set with potentially false comparison answers, providing improved algorithms and bounds for the number of comparisons needed when up to k lies are allowed.

Contribution

The paper introduces a new algorithm that reduces the number of comparisons needed to find min and max with up to k lies, improving previous bounds.

Findings

New algorithm with at most (k+1+C)n+O(k^3) comparisons

Improved upper bounds for comparison complexity with lies

Refined lower bounds showing asymptotic behavior of comparison counts

Abstract

A neat 1972 result of Pohl asserts that [3n/2]-2 comparisons are sufficient, and also necessary in the worst case, for finding both the minimum and the maximum of an n-element totally ordered set. The set is accessed via an oracle for pairwise comparisons. More recently, the problem has been studied in the context of the Renyi-Ulam liar games, where the oracle may give up to k false answers. For large k, an upper bound due to Aigner shows that (k+O(\sqrt{k}))n comparisons suffice. We improve on this by providing an algorithm with at most (k+1+C)n+O(k^3) comparisons for some constant C. The known lower bounds are of the form (k+1+c_k)n-D, for some constant D, where c_0=0.5, c_1=23/32=0.71875, and c_k=\Omega(2^{-5k/4}) as k goes to infinity.