On selecting the largest element in spite of erroneous information
4th Annual Symposium on Theoretical Aspects of Computer Sciences on STACS 87
Finding the maximum and minimum
Discrete Applied Mathematics
SIAM Journal on Computing
A sorting problem and its complexity
Communications of the ACM
Searching games with errors---fifty years of coping with liars
Theoretical Computer Science
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Finding the maximum and minimum elements with one lie
Discrete Applied Mathematics
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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+ck)n−D, for some constant D, where c0=0.5, $c_1=\frac{23}{32}= 0.71875$, and $c_k={\mathrm{\Omega}}(2^{-5k/4})$ as k→∞.