{"title":"整数反馈组合查询问题的最优方案","authors":"Anders Martinsson","doi":"10.5070/c63261997","DOIUrl":null,"url":null,"abstract":"A query game is a pair of a set \\(Q\\) of queries and a set \\(\\mathcal{F}\\) of functions, or codewords \\(f:Q\\rightarrow \\mathbb{Z}.\\) We think of this as a two-player game. One player, Codemaker, picks a hidden codeword \\(f\\in \\mathcal{F}\\). The other player, Codebreaker, then tries to determine \\(f\\) by asking a sequence of queries \\(q\\in Q\\), after each of which Codemaker must respond with the value \\(f(q)\\). The goal of Codebreaker is to uniquely determine \\(f\\) using as few queries as possible. Two classical examples of such games are coin-weighing with a spring scale, and Mastermind, which are of interest both as recreational games and for their connection to information theory.In this paper, we will present a general framework for finding short solutions to query games. As applications, we give new self-contained proofs of the query complexity of variations of the coin-weighing problems, and prove new results that the deterministic query complexity of Mastermind with \\(n\\) positions and \\(k\\) colors is \\(\\Theta(n \\log k/ \\log n + k)\\) if only black-peg information is provided, and \\(\\Theta(n \\log k / \\log n + k/n)\\) if both black- and white-peg information is provided. In the deterministic setting, these are the first up to constant factor optimal solutions to Mastermind known for any \\(k\\geq n^{1-o(1)}\\).Mathematics Subject Classifications: 91A46, 68Q11, 05B99Keywords: Combinatorial games, query complexity, Mastermind, coin-weighing","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Optimal schemes for combinatorial query problems with integer feedback\",\"authors\":\"Anders Martinsson\",\"doi\":\"10.5070/c63261997\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A query game is a pair of a set \\\\(Q\\\\) of queries and a set \\\\(\\\\mathcal{F}\\\\) of functions, or codewords \\\\(f:Q\\\\rightarrow \\\\mathbb{Z}.\\\\) We think of this as a two-player game. One player, Codemaker, picks a hidden codeword \\\\(f\\\\in \\\\mathcal{F}\\\\). The other player, Codebreaker, then tries to determine \\\\(f\\\\) by asking a sequence of queries \\\\(q\\\\in Q\\\\), after each of which Codemaker must respond with the value \\\\(f(q)\\\\). The goal of Codebreaker is to uniquely determine \\\\(f\\\\) using as few queries as possible. Two classical examples of such games are coin-weighing with a spring scale, and Mastermind, which are of interest both as recreational games and for their connection to information theory.In this paper, we will present a general framework for finding short solutions to query games. As applications, we give new self-contained proofs of the query complexity of variations of the coin-weighing problems, and prove new results that the deterministic query complexity of Mastermind with \\\\(n\\\\) positions and \\\\(k\\\\) colors is \\\\(\\\\Theta(n \\\\log k/ \\\\log n + k)\\\\) if only black-peg information is provided, and \\\\(\\\\Theta(n \\\\log k / \\\\log n + k/n)\\\\) if both black- and white-peg information is provided. In the deterministic setting, these are the first up to constant factor optimal solutions to Mastermind known for any \\\\(k\\\\geq n^{1-o(1)}\\\\).Mathematics Subject Classifications: 91A46, 68Q11, 05B99Keywords: Combinatorial games, query complexity, Mastermind, coin-weighing\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2023-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.5070/c63261997\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5070/c63261997","RegionNum":2,"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
摘要
查询博弈是一组\(Q\)的查询和一组\(\mathcal{F}\)的函数,或者代码字\(f:Q\rightarrow \mathbb{Z}.\),我们认为这是一个双人博弈。一个玩家,Codemaker,选择一个隐藏的码字\(f\in \mathcal{F}\)。另一个玩家,Codebreaker,然后尝试通过查询一系列\(q\in Q\)来确定\(f\),在每个查询之后,Codemaker都必须响应值\(f(q)\)。Codebreaker的目标是使用尽可能少的查询来唯一地确定\(f\)。这类游戏的两个经典例子是用弹簧秤称重硬币和《Mastermind》,这两款游戏不仅是娱乐游戏,而且与信息理论有关。在本文中,我们将呈现一个寻找查询游戏的简短解决方案的一般框架。作为应用,我们给出了硬币称重问题变体查询复杂度的新自包含证明,并证明了具有\(n\)位置和\(k\)颜色的Mastermind在只提供黑钉信息时的确定性查询复杂度为\(\Theta(n \log k/ \log n + k)\),在同时提供黑钉和白钉信息时的确定性查询复杂度为\(\Theta(n \log k / \log n + k/n)\)。在确定性设置中,这些是已知的任何\(k\geq n^{1-o(1)}\) .数学学科分类:91A46, 68Q11, 05b99关键字:组合游戏,查询复杂性,Mastermind,硬币称重
Optimal schemes for combinatorial query problems with integer feedback
A query game is a pair of a set \(Q\) of queries and a set \(\mathcal{F}\) of functions, or codewords \(f:Q\rightarrow \mathbb{Z}.\) We think of this as a two-player game. One player, Codemaker, picks a hidden codeword \(f\in \mathcal{F}\). The other player, Codebreaker, then tries to determine \(f\) by asking a sequence of queries \(q\in Q\), after each of which Codemaker must respond with the value \(f(q)\). The goal of Codebreaker is to uniquely determine \(f\) using as few queries as possible. Two classical examples of such games are coin-weighing with a spring scale, and Mastermind, which are of interest both as recreational games and for their connection to information theory.In this paper, we will present a general framework for finding short solutions to query games. As applications, we give new self-contained proofs of the query complexity of variations of the coin-weighing problems, and prove new results that the deterministic query complexity of Mastermind with \(n\) positions and \(k\) colors is \(\Theta(n \log k/ \log n + k)\) if only black-peg information is provided, and \(\Theta(n \log k / \log n + k/n)\) if both black- and white-peg information is provided. In the deterministic setting, these are the first up to constant factor optimal solutions to Mastermind known for any \(k\geq n^{1-o(1)}\).Mathematics Subject Classifications: 91A46, 68Q11, 05B99Keywords: Combinatorial games, query complexity, Mastermind, coin-weighing
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.