{"title":"Ranking and Unranking k-subsequence universal words","authors":"Duncan Adamson","doi":"10.48550/arXiv.2304.04583","DOIUrl":null,"url":null,"abstract":"A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] , \\dots w[i_{|u|}]$, for some set of indices $1 \\leq i_1<i_2<\\dots<i_k \\leq |w|$. A word $w$ is $k$-subsequence universal over an alphabet $\\Sigma$ if every word in $\\Sigma^k$ appears in $w$ as a subsequence. In this paper, we provide new algorithms for $k$-subsequence universal words of fixed length $n$ over the alphabet $\\Sigma = \\{1,2,\\dots, \\sigma\\}$. Letting $\\mathcal{U}(n,k,\\sigma)$ denote the set of $n$-length $k$-subsequence universal words over $\\Sigma$, we provide: * an $O(n k \\sigma)$ time algorithm for counting the size of $\\mathcal{U}(n,k,\\sigma)$; * an $O(n k \\sigma)$ time algorithm for ranking words in the set $\\mathcal{U}(n,k,\\sigma)$; * an $O(n k \\sigma)$ time algorithm for unranking words from the set $\\mathcal{U}(n,k,\\sigma)$; * an algorithm for enumerating the set $\\mathcal{U}(n,k,\\sigma)$ with $O(n \\sigma)$ delay after $O(n k \\sigma)$ preprocessing.","PeriodicalId":31852,"journal":{"name":"Beyond Words","volume":"114 1","pages":"47-59"},"PeriodicalIF":0.0000,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Beyond Words","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2304.04583","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 3
Abstract
A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] , \dots w[i_{|u|}]$, for some set of indices $1 \leq i_1
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