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arXiv · 2003.11538

The Exact Query Complexity of Yes-No Permutation Mastermind

Moura El Ouali·Volkmar Sauerland
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Abstract

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 $k\le n$ colors we prove a lower bound of $\log_2(k+1-n)+\log_2(k+2-n)+\dots+\log_2(k)$ 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 $\Theta(n\log_2(n))$ queries.

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Moura El Ouali, Volkmar Sauerland. 2020-03-25. The Exact Query Complexity of Yes-No Permutation Mastermind. https://arxiv.org/abs/2003.11538

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