Minimum and maximum against k lies
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.
Keywords
Cite
@article{arxiv.1002.0562,
title = {Minimum and maximum against k lies},
author = {Michael Hoffmann and Jiří Matoušek and Yoshio Okamoto and Philipp Zumstein},
journal= {arXiv preprint arXiv:1002.0562},
year = {2015}
}
Comments
11 pages, 3 figures