Advertisement

Search When the Lie Depends on the Target

  • Gyula O. H. Katona
  • Krisztián Tichler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7777)

Abstract

The following model is considered. There is exactly one unknown element in the n-element set. A question is a partition of S into three classes: (A,L,B). If x ∈ A then the answer is “yes” (or 1), if x ∈ B then the answer is “no” (or 0), finally if x ∈ L then the answer can be either “yes” or “no”. In other words, if the answer “yes” is obtained then we know that x ∈ A ∪ L while in the case of “no” answer the conclusion is x ∈ B ∪ L. The mathematical problem is to minimize the minimum number of questions under certain assumptions on the sizes of A,B and L. This problem has been solved under the condition |L| ≥ k by the author and Krisztián Tichler in previous papers for both the adaptive and non-adaptive cases. In this paper we suggest to solve the problem under the conditions |A| ≤ a, |B| ≤ b. We exhibit some partial results for both the adaptive and non-adaptive cases. We also show that the problem is closely related to some known combinatorial problems. Let us mention that the case b = n − a has been more or less solved in earlier papers.

Keywords

combinatorial search search with lies 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlswede, R.: General theory of information transfer: updated, General Theory of Information Transfer and Combinatorics. Special Issue of Discrete Applied Mathematics 156(92), 1348–1388 (2008), http://www.math.uni-bielefeld.de/ahlswede/homepage/public/220.pdfCrossRefzbMATHGoogle Scholar
  2. 2.
    Ahlswede, R., Wegener, I.: Search Problems. Wiley Interscience Series in Discrete Mathematics. John Wiley & Sons Inc. (1980)Google Scholar
  3. 3.
    Aigner, M.: Combinatorial Search. John Wiley & Sons, Inc., New York (1988)zbMATHGoogle Scholar
  4. 4.
    Bassalygo, L., Kabatianski, G.: Personal communicationGoogle Scholar
  5. 5.
    Berlekamp, E.R.: Block coding for the binary symmetric channel with noiseless, delayless feedback. In: Mann, H.B. (ed.) Error Correcting Codes. Wiley (1968)Google Scholar
  6. 6.
    Berlekamp, E.R., Hill, R., Karim, J.: The solution of a problem of Ulam on searching with lies. In: IEEE Int. Symp. on Inf. Theory, vol. 244. MIT, Cambridge (1998)Google Scholar
  7. 7.
    Bollobás, B., Scott, A.: On separating systems. European J. Combin. 28(4), 1068–1071 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bollobás, B., Scott, A.: Separating systems and oriented graphs of diameter two. J. Combin. Theory Ser. B 97(2), 193–203 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Deppe, C.: Coding with feedback and searching with lies. In: Csiszár, I., Katona, G.O.H., Tardos, G. (eds.) Entropy, Search, Complexity. Bolyai Society Mathematical Studies, vol. 16, pp. 27–70 (2007)Google Scholar
  10. 10.
    Du, D.-Z., Hwang, F.K.: Combinatorial Group Testing. World Scientific (1993)Google Scholar
  11. 11.
    Katona, G.: Combinatorial Search Problems. In: Srivastava, J.N. (ed.) A Survey of Combinatorial Theory, pp. 285–308. North Holland/American Elsevier, Amsterdam/New York (1973)CrossRefGoogle Scholar
  12. 12.
    Katona, G.O.H.: Search with small sets in presence of a liar. J. Statictical Planning and Inference 100, 319–336 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Katona, G., Szemerédi, E.: On a problem of graph theory. Studia Sci. Math. Hungar. 2, 23–28 (1967)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Katona, G.O.H., Tichler, K.: Existence of a balanced matching in the hypercube (submitted)Google Scholar
  15. 15.
    Pálvölgyi, D.: Personal communicationGoogle Scholar
  16. 16.
    Rényi, A.: On a problem of information theory. MTA Mat. Int. Közl., 6B, 505–516 (1961)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ulam, S.: Adventures of a Mathematician. Scribner, New York (1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gyula O. H. Katona
    • 1
  • Krisztián Tichler
    • 2
  1. 1.Rényi InstituteBudapestHungary
  2. 2.Eötvös UniversityBudapestHungary

Personalised recommendations