Solution of Ulam's problem on searching with a lie
Journal of Combinatorial Theory Series A
Ulam's searching game with a fixed number of lies
Theoretical Computer Science
Surveys in combinatorics, 1995
Optimal strategies against a liar
Theoretical Computer Science
Optimal Binary Search with Two Unreliable Tests and Minimum Adaptiveness
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
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Ulam and RÉnyi asked what is the minimum number of yes-no questions needed to find an unknown m-bit number x, if up to l of the answers may be erroneous/mendacious. For each l it is known that, up to only finitely many exceptional m, one can find x asking Berlekamp's minimum number ql(m) of questions, i.e., the smallest integer q satisfying the sphere packing bound for error-correcting codes. The Ulam-RÉnyi problem amounts to finding optimal error-correcting codes for the binary symmetric channel with noiseless feedback, first considered by Berlekamp. In such concrete situations as optical transmission, error patterns are highly asymmetric--in that only one of the two bits can be distorted. Optimal error-correcting codes for these asymmetric channels with feedback are the solutions of the half-lie variant of the Ulam-RÉnyi problem, asking for the minimum number of yes-no questions needed to find an unknown m-bit number x, if up to l of the negative answers may be erroneous/mendacious. Focusing attention on the case l = 1; in this self-contained paper we shall give tight upper and lower bounds for the half-lie problem. For infinitely many m's our bounds turn out to be matching, and the optimal solution is explicitly given, thus strengthening previous estimates by Rivest, Meyer et al.