Mastermind变体的查询复杂度
Query Complexity of Mastermind Variants
摘要：
我们研究的变体的主谋，一个流行的棋盘游戏，其目标是序列重建。在这种二人游戏中，所谓的\textit{codemaker}构造从字母表$\mathcal{A} = \{1,2,\ldots, k\}$中选择的颜色的隐藏序列$H = (h_1, h_2, \ldots, h_n)$（对于所有$i\in\{1,2,\ldots, n\}$，\textit{i.e.,} $h_i\in\mathcal{A}$）。然后游戏依次进行，每个部分由两部分组成：依次是$t$，第二个玩家（\textit{codebreaker}）首先提交一个查询序列$Q_t = (q_1, q_2, \ldots, q_n)$，其中所有$i$都有$q_i\in \mathcal{A}$，第二个玩家收到反馈$Δ(Q_t, H)$，其中$Δ$是两个序列之间的距离的某个商定函数，其中$n$是两个序列之间的距离。当$Q_t = H$出现时，游戏结束，而密码破译者会尽可能地在最少的回合内结束游戏。在整个过程中，我们让$f(n,k)$表示最小的整数，使得密码破译器可以确定$f(n,k)$轮次中的任何$H$。我们证明了三个主要结果：首先，当$H$是已知的$\{1,2,\ldots, n\}$的置换，我们证明了$f(n, n)\ge n - \log\log n$对所有足够大的$n$。第二，我们表明，克努特的Minimax算法识别任何$H$在大多数$nk$查询。第三，当在提交所有查询之前没有收到反馈时，我们表明$f(n,k)=Ω(n\log k)$。
英文摘要：
We study variants of Mastermind, a popular board game in which the objective is sequence reconstruction. In this two-player game, the so-called \textit{codemaker} constructs a hidden sequence $H = (h_1, h_2, \ldots, h_n)$ of colors selected from an alphabet $\mathcal{A} = \{1,2,\ldots, k\}$ (\textit{i.e.,} $h_i\in\mathcal{A}$ for all $i\in\{1,2,\ldots, n\}$). The game then proceeds in turns, each of which consists of two parts: in turn $t$, the second player (the \textit{codebreaker}) first submits a query sequence $Q_t = (q_1, q_2, \ldots, q_n)$ with $q_i\in \mathcal{A}$ for all $i$, and second receives feedback $Δ(Q_t, H)$, where $Δ$ is some agreed-upon function of distance between two sequences with $n$ components. The game terminates when $Q_t = H$, and the codebreaker seeks to end the game in as few turns as possible. Throughout we let $f(n,k)$ denote the smallest integer such that the codebreaker can determine any $H$ in $f(n,k)$ turns. We prove three main results: First, when $H$ is known to be a permutation of $\{1,2,\ldots, n\}$, we prove that $f(n, n)\ge n - \log\log n$ for all sufficiently large $n$. Second, we show that Knuth's Minimax algorithm identifies any $H$ in at most $nk$ queries. Third, when feedback is not received until all queries have been submitted, we show that $f(n,k)=Ω(n\log k)$.
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。
