Minimum and Maximum against k Lies

  • Michael Hoffmann
  • Jiří Matoušek
  • Yoshio Okamoto
  • Philipp Zumstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

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→ ∞.

Keywords

Generic Algorithm Simple Algorithm Directed Edge Small Element Large Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michael Hoffmann
    • 1
  • Jiří Matoušek
    • 2
    • 1
  • Yoshio Okamoto
    • 3
  • Philipp Zumstein
    • 1
  1. 1.Institute of Theoretical Computer Science, Department of Computer ScienceETH ZurichSwitzerland
  2. 2.Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic
  3. 3.Graduate School of Information Science and EngineeringTokyo Institute of TechnologyJapan

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