Solution of Ulam's problem on searching with a lie
Journal of Combinatorial Theory Series A
Combinatorial search
Ulam's searching game with lies
Journal of Combinatorial Theory Series A
Searching in the presence of linearly bounded errors
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Ulam's searching game with a fixed number of lies
Theoretical Computer Science
On playing “Twenty Questions” with a liar
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
Ulam's searching game with three lies
Advances in Applied Mathematics
Comparison-based search in the presence of errors
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Surveys in combinatorics, 1995
Journal of Combinatorial Theory Series A
On-line prediction and conversion strategies
Machine Learning
Optimal comparison strategies in Ulam's searching game with two errors
Theoretical Computer Science
On optimal strategies for searching in presence of errors
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
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
Protocols for Asymmetric Communication Channels
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Coping with errors in binary search procedures (Preliminary Report)
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
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We consider the basic problem of searching for an unknown m-bit number by asking the minimum possible number of yes-no questions, when up to a finite number e of the answers may be erroneous. In case the (i + 1)th question is adaptively asked after receiving the answer to the ith question, the problem was posed by Ulam and RÉnyi and is strictly related to Berlekamp's theory of error correcting communication with noiseless feedback. Conversely, in the fully non-adaptive model when all questions are asked before knowing any answer, the problem amounts to finding a shortest e-error correcting code. Let qe(m) be the smallest integer q satisfying Berlekamp's bound Σi=0e (iq) ≤ 2q-m. Then at least qe(m) questions are necessary, in the adaptive, as well as in the non-adaptive model. In the fully adaptive case, optimal searching strategies using exactly qe(m) questions always exist up to finitely many exceptional m's. At the opposite non-adaptive case, searching strategies with exactly qe(m) questions--or equivalently, perfect e-error correcting codes with 2m codewords of length qe(m)--are rather the exception, already for e = 2, and do not exist for e 2. In this paper we show that for any e 1 and sufficiently large m, optimal--indeed, perfect-- strategies do exist using a first batch of m non-adaptive questions and then, only depending on the answers to these m questions, a second batch of qe(m)-m non-adaptive questions. Since even in the fully adaptive case, qe(m)-1 questions do not suffice to find the unknown number, and qe(m) questions generally do not suffice in the non-adaptive case, the results of our paper provide e-fault tolerant searching strategies with minimum adaptiveness and minimum number of tests.