{"title":"Optimal Sets of Questions for Twenty Questions","authors":"Yuval Filmus, Idan Mehalel","doi":"10.1137/21m1424494","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 412-452, March 2024. <br/> 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].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/21m1424494","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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].
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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