Minimum and maximum against k lies

Jiří Matoušek

Abstract

(joint work with Michael Hoffmann, Yoshio Okamoto, and Philipp Zumstein)

A neat 1972 result of Pohl asserts that 3n/2+O(1) 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, and c_k=Omega(2^{-5k/4}) for large k.