k-子序列全称词排序与取消排序

Beyond Words Pub Date : 2023-04-10 DOI:10.48550/arXiv.2304.04583

Duncan Adamson

{"title":"k-子序列全称词排序与取消排序","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":"{\"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}","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}