The Exact Query Complexity of Yes-No Permutation Mastermind
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Mastermind is famous two-players game. The first player (codemaker) chooses a secret code which the second player (codebreaker) is supposed to crack within a minimum number of code guesses (queries). Therefore, codemaker's duty is to help codebreaker by providing a well-defined error measure between the secret code and the guessed code after each query. We consider a variant, called Yes-No AB-Mastermind, where both secret code and queries must be repetition-free and the provided information by codemaker only indicates if a query contains any correct position at all. For this Mastermind version with n positions and k< n colors we prove a lower bound of log_2(k+1-n)+log_2(k+2-n)+...+log_2(k) and an upper bound of nlog_2(n)+k on the number of queries necessary to break the secret code. For the important case k=n, where both secret code and queries represent permutations, our results imply an exact asymptotic complexity of Θ(nlog_2(n)) queries.
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