Solving Static Permutation Mastermind using $O(n \log n)$ Queries

The Exact Query Complexity of Yes-No Permutation Mastermind

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…

线性查询次数的珠玑妙算

Mastermind with a Linear Number of Queries

本文针对k=n时珠玑妙算查询复杂度的开放问题，提出首个线性查询次数算法，从而确定任意参数k和n下的查询复杂度。

局部性在排列大师游戏中定价：TikTok影响者够混沌吗？

Price of Locality in Permutation Mastermind: Are TikTok influencers Chaotic Enough?

研究局部策略在排列大师游戏中的影响，显示局部策略需要二次猜测，远逊于非局部策略的O(n lg n)。

The Query Complexity of Mastermind with $\ell_p$ Distances

Consider a variant of the Mastermind game in which queries are $\ell_p$ distances, rather than the usual Hamming distance. That is, a codemaker chooses a hidden vector $\mathbf{y}\in\{-k,-k+1,\dots,k-1,k\}^n$ and answers to queries of the form $\Vert\mathbf{y}-\mathbf{x}\Vert_p$ where $\mathbf{x}\in\{-k,-k+1,\dots,k-1,k\}^n$. The goal is to minimize the number of queries made in order to correctly guess $\mathbf{y}$. Motivated by this question, in this work, we develop a nonadaptive polynomial time algorithm that works for a natural class of separable distance measures, i.e.\ coordinate-wise s…

Mastermind变体的查询复杂度

Query Complexity of Mastermind Variants

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…

具有整数反馈的组合查询问题的最优方案

Optimal schemes for combinatorial query problems with integer feedback

本文提出查询游戏短解的一般框架，并据此给出硬币称重变体的新证明及 Mastermind 在两种反馈信息下的确定性查询复杂度最优界。