Search When the Lie Depends on the Target
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.
Keywordscombinatorial search search with lies
Unable to display preview. Download preview PDF.
- 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.pdfCrossRefMATHGoogle Scholar
- 2.Ahlswede, R., Wegener, I.: Search Problems. Wiley Interscience Series in Discrete Mathematics. John Wiley & Sons Inc. (1980)Google Scholar
- 4.Bassalygo, L., Kabatianski, G.: Personal communicationGoogle Scholar
- 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.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
- 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.Du, D.-Z., Hwang, F.K.: Combinatorial Group Testing. World Scientific (1993)Google Scholar
- 14.Katona, G.O.H., Tichler, K.: Existence of a balanced matching in the hypercube (submitted)Google Scholar
- 15.Pálvölgyi, D.: Personal communicationGoogle Scholar
- 17.Ulam, S.: Adventures of a Mathematician. Scribner, New York (1976)Google Scholar