Published by the Department of Computer Science, The University of Chicago.

Abstract

A neat 1972 result of Pohl asserts that ceiling(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+BigO(sqrt{k}))n comparisons suffice. We improve on this by providing an algorithm with at most (k+1+C)n+BigO(k^3) comparisons for some constant C. A recent result of Pálvölgyi provides a lower bound of the form (k+1+0.5)n-D_k, so our upper bound for the coefficient of n is tight up to the value of C.

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Submitted November 26,2010, revised June 20, 2012, 2, published July 1, 2012.

© 2012 Michael Hoffman, Jiŕí Matoušek, Yoshio Okamoto and Philipp Zumstein
Licensed under a Creative Commons Attribution License

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