线性查询次数的珠玑妙算
Mastermind with a Linear Number of Queries
AI总结：
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。
摘要：
自20世纪60年代以来，珠玑妙算因其组合与信息论方面的趣味性而受到研究。许多结果相继被发现，最早可追溯到Erdős和Rényi确定了两种颜色所需的最优查询次数。对于k种颜色和n个位置，Chvátal在k ≤ n^{1-ε}时找到了渐近最优界。随着k ≥ n种颜色情形的一系列逐步改进，核心的开放问题是解决k=n时Ω(n)与O(n log log n)之间的差距。本文通过提出第一个以线性查询次数求解k=n珠玑妙算的算法，弥合了这一差距。由此，我们能够确定任意参数k和n下珠玑妙算的查询复杂度。
英文摘要：
Since the 1960s Mastermind has been studied for the combinatorial and information theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colors. For $k$ colors and $n$ positions, Chvátal found asymptotically optimal bounds when $k \le n^{1-ε}$. Following a sequence of gradual improvements for $k \geq n$ colors, the central open question is to resolve the gap between $Ω(n)$ and $\mathcal{O}(n\log \log n)$ for $k=n$. In this paper, we resolve this gap by presenting the first algorithm for solving $k=n$ Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters $k$ and $n$.
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。
