Yes-No置换摄护仪的精确查询复杂度
The Exact Query Complexity of Yes-No Permutation Mastermind
摘要：
主谋是著名的两个球员的游戏。第一个玩家（代码制作者）选择一个密码，第二个玩家（代码破解者）应该在最小数量的代码猜测（查询）内破解该密码。因此，编码者的职责是通过在每次查询之后提供秘密代码和猜测代码之间的定义良好的误差度量来帮助解码者。我们考虑一个变种，称为是-否AB-Mastermind，其中密码和查询都必须是无重复的，并且由codemaker提供的信息仅指示查询是否包含任何正确的位置。对于这一版本的Mastermind与n个位置和$k\le n$的颜色，我们证明了一个下界的$\log_2(k+1-n)+\log_2(k+2-n)+\dots+\log_2(k)$和一个上限的$n\log_2(n)+k$上的数量的查询，需要打破的秘密代码。对于重要的情况下$k=n$，密码和查询表示置换，我们的结果意味着一个确切的渐近复杂性的$Θ(n\log_2(n))$查询。
英文摘要：
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 $Θ(n\log_2(n))$ queries.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
