Searching games with errors---fifty years of coping with liars
Theoretical Computer Science
Least adaptive optimal search with unreliable tests
Theoretical Computer Science
The multi-interval ulam-rényi game
FUN'12 Proceedings of the 6th international conference on Fun with Algorithms
| Hi-index | 0.01 |
Our aim is to minimize the number of answer/question alternations in a perfect two-fault-tolerant search. Ulam and Renyi posed the problem of searching for an unknown m-bit number by asking the minimum number of yes-no questions, when up to @? of the answers may be erroneous/mendacious. Berlekamp considered the same problem in the context of error-correcting communication with feedback. Among others, he proved that at least q"@?(m) questions are necessary, where q"@?(m) is the smallest integer q satisfying 2^q^-^m=@?^@?"j"="0(^q"j). When all questions are asked in advance, and adaptiveness has no role, finding a perfect strategy (i.e., a strategy with q"@?(m) questions) amounts to finding an @?-error-correcting code with 2^m codewords of length q"@?(m). From coding theory it is known that such perfect non-adaptive searching strategies are rather the exception, for @?=2. At the other extreme, in a fully adaptive search, where the (t+1)th question is asked knowing the answer to the tth question, perfect strategies are known to exist for all sufficiently large m. What happens if we impose restrictions on the amount of adaptiveness avaliable to the questioner? Focusing attention on the case @?=2, we shall prove that, for each m2, perfect searching strategies still exist even if the questioner is allowed to adapt his strategy only once. All our results are constructive and explicitly yield perfect two-error-correcting codes with the least possible feedback. We finally generalize our results to k-ary search.