Optimal Sets of Questions for Twenty Questions

IF 0.9 3区 数学 Q2 MATHEMATICS
Yuval Filmus, Idan Mehalel
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引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 412-452, March 2024.
Abstract. In the distributional Twenty Questions game, Bob chooses a number [math] from 1 to [math] according to a distribution [math], and Alice (who knows [math]) attempts to identify [math] using yes/no questions, which Bob answers truthfully. Her goal is to minimize the expected number of questions. The optimal strategy for the Twenty Questions game corresponds to a Huffman code for [math], yet this strategy could potentially uses all [math] possible questions. Dagan et al. constructed a set of [math] questions which suffice to construct an optimal strategy for all [math], and showed that this number is optimal (up to subexponential factors) for infinitely many [math]. We determine the optimal size of such a set of questions for all [math] (up to subexponential factors), answering an open question of Dagan et al. In addition, we generalize the results of Dagan et al. to the [math]-ary setting, obtaining similar results with 1.25 replaced by [math].
二十个问题的最佳问题集
SIAM 离散数学杂志》,第 38 卷,第 1 期,第 412-452 页,2024 年 3 月。 摘要。在分布式二十问博弈中,鲍勃根据分布式[math]从 1 到[math]中选择一个数字[math],而爱丽丝(知道[math])试图用是/否问题来识别[math],鲍勃则如实回答。她的目标是最小化预期问题数。二十个问题游戏的最优策略对应于[数学]的哈夫曼编码,但这一策略有可能使用所有[数学]可能的问题。达根等人构建了一组[数学]问题,足以为所有[数学]构建一个最优策略,并证明这个数量对于无限多的[数学]来说是最优的(达到亚指数因子)。我们确定了这样一组问题对所有 [math] 的最优规模(达到亚指数因子),回答了达甘等人的一个公开问题。此外,我们将达甘等人的结果推广到 [math]ary 环境,得到了用 [math] 代替 1.25 的类似结果。
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来源期刊
CiteScore
1.90
自引率
0.00%
发文量
124
审稿时长
4-8 weeks
期刊介绍: SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.
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