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
Weakly adaptive comparison searching
Theoretical Computer Science
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
Regular Article: Perfect Two-Fault Tolerant Search with Minimum Adaptiveness
Advances in Applied Mathematics
Optimal Binary Search with Two Unreliable Tests and Minimum Adaptiveness
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Coping with errors in binary search procedures (Preliminary Report)
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Strategies for the Renyi-Ulam game with fixed number of lies
Theoretical Computer Science
Searching with lies under error cost constraints
Discrete Applied Mathematics
Two batch search with lie cost
IEEE Transactions on Information Theory
Q-Ary ulam-renyi game with constrained lies
General Theory of Information Transfer and Combinatorics
The multi-interval ulam-rényi game
FUN'12 Proceedings of the 6th international conference on Fun with Algorithms
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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 R&ényi 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 i=0e()2qm. 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, e-error correcting codes with 2m codewords of length qe(m)---are rather the exception, already for e=2, and are generally not known to exist for e2. In this paper, for each e1 and all sufficiently large m, we exhibit searching strategies that use 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. These strategies are automatically optimal. 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.