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 Rényi–Ulam liar games, where the oracle may give up to k false answers. For large k, an upper bound due to Aigner shows that $(k+{\mathcal O}(\sqrt{k}))n$ comparisons suffice. We improve on this by providing an algorithm with at most $(k+1+C)n+{\mathcal O}(k^3)$ 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, $c_1=\frac{23}{32}= 0.71875$, and $c_k={\mathrm{\Omega}}(2^{-5k/4})$ as k→ ∞.