The Exact Query Complexity of Yes-No Permutation Mastermind

Mastermind is famous two-player 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, the codemaker’s duty is to help the 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 the codemaker only indicates if a query contains any correct position at all. For this Mastermind version with n positions and k ≥ n colors and ` := k + 1 − n, we prove a lower bound of ∑ k j=` log 2 j and an upper bound of n log 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 (n log n) queries.


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