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Rényi-Berlekamp-Ulam searching game with bi-interval queries and two lies. (English) Zbl 1338.91041
Summary: We consider the following searching game: there are two players, say Questioner and Responder. Responder chooses a number $$x\in S_n=\{1,2, \ldots,n\}$$, Questioner has to find out the number $$x$$ by asking bi-interval queries and Responder is allowed to lie at most two times throughout the game. The minimal number $$q^\ast(n)$$ of bi-interval queries sufficient to find the unknown integer $$x$$ is determined for all integers $$n$$. This solves completely Rényi-Berlekamp-Ulam searching game with bi-interval queries and two lies, partially solved by D. Mundici and A. Trombetta [Theor. Comput. Sci. 182, No. 1–2, 217–232 (1997; Zbl 0902.90191)]. Their solution applied only to the case when $$n$$ is a power of 2.
##### MSC:
 91A46 Combinatorial games 68P10 Searching and sorting 91A05 2-person games
##### Keywords:
Rényi-Ulam game; search; Lie; bi-interval queries; worst-case
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##### References:
 [1] Aigner, M., Searching with lies, J. Combin. Theory Ser. A, 74, 43-56, (1996) · Zbl 0846.90149 [2] Berlekamp, E. R., Block coding for the binary symmetric channel with feedback, (Mann, H. B., Error-correcting Codes, (1968), Wiley New York), 330-335 [3] Cicalese, F., Fault-tolerant search algorithms, (2013), Springer-Verlag · Zbl 1295.68006 [4] Cicalese, F., Perfect strategies for the Ulam-Rényi game with multi-interval questions, Theory Comput. Syst., 54, 4, 578-594, (2014) · Zbl 1303.68044 [5] Cicalese, F.; Mundici, D.; Vaccaro, U., Rota-metropolis cubic logic and Ulam-Rényi games, (Crapo, H.; Senato, D., Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota, (2001), Spinger-Verlag), 197-244 · Zbl 0978.03045 [6] Cicalese, F.; Vaccaro, U., Optimal strategies against a liar, Theoret. Comput. Sci., 230, 167-193, (2000) · Zbl 0966.91014 [7] Czyzowicz, J.; Mundici, D.; Pelc, A., Solution of ulam’s problem on binary search with two lies, J. Combin. Theory Ser. A, 49, 384-388, (1998) · Zbl 0662.68059 [8] Czyzowicz, J.; Mundici, D.; Pelc, A., Ulam’s searching game with lies, J. Combin. Theory Ser. A, 52, 62-76, (1989) · Zbl 0674.90110 [9] Deppe, C., Solution of ulam’s searching game with three lies or an optimal adaptive strategy for binary three-error-correcting codes, Discrete Math., 224, 79-98, (2000) · Zbl 0967.91008 [10] Deppe, C., Coding with feedback and searching with lies, (Entropy, Search, Complexity, (2007), Springer-Verlag), 27-70 [11] Ellis, R. B.; Ponomarenko, Vadim; Yan, C. H., The renyi-Ulam pathological liar game with a fixed number of lies, J. Combin. Theory Ser. A, 112, 2, 328-336, (2005) · Zbl 1121.91023 [12] Guzicki, W., Ulam’s searching game with two lies, J. Combin. Theory Ser. A, 54, 1-19, (1990) · Zbl 0712.68027 [13] Hill, R., Searching with lies, (Rowlinson, P., Surveys in Combinatorics, Lecture Note Series, vol. 218, (1995), Cambridge Univ. Press Cambridge), 41-70 · Zbl 0833.90129 [14] Liu, W. A.; Zhang, Q. M.; Nie, Z. K., Searching for a counterfeit coin with two unreliable weighings, Discrete Appl. Math., 150, 160-181, (2005) · Zbl 1071.05003 [15] Malinowski, A., K-ary searching with a Lie, Ars Combin., 37, 301-308, (1994) · Zbl 0820.68039 [16] Mundici, D.; Trombetta, A., Optimal comparison strategies in ulam’s searching game with two errors, Theoret. Comput. Sci., 182, 217-232, (1997) · Zbl 0902.90191 [17] Negro, A.; Sereno, M., Ulam’s searching game with three lies, Adv. Appl. Math., 13, 404-428, (1992) · Zbl 0773.68022 [18] Negro, A.; Sereno, M., Solution of ulam’s problem on binary search with three lies, J. Combin. Theory Ser. A, 59, 149-154, (1992) · Zbl 0763.05004 [19] Pelc, A., Solution of ulam’s problem on searching with a Lie, J. Combin. Theory Ser. A, 44, 129-140, (1987) · Zbl 0621.68056 [20] Pelc, A., Searching games with errors-fifty years of coping with liars, Theoret. Comput. Sci., 270, 71-109, (2002) · Zbl 0984.68041 [21] Pelc, A., Detecting a counterfeit coin with unreliable weighings, Ars Combin., 27, 181-192, (1989) · Zbl 0719.05008 [22] Rényi, A., On a problem of information theory, MTA Mat. Kut. Int. Kozl., 6B, 505-516, (1961) · Zbl 0126.35605 [23] Spencer, J., Ulam’s searching game with a fixed number of lies, Theoret. Comput. Sci., 95, 2, 307-321, (1992) · Zbl 0749.90102 [24] Ulam, S. M., Adventures of a Mathematician, (1976), Scribner’s New York · Zbl 0352.01009
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