{"title":"Algorithms with improved delay for enumerating connected induced subgraphs of a large cardinality","authors":"Shanshan Wang, Chenglong Xiao, E. Casseau","doi":"10.2139/ssrn.4150167","DOIUrl":null,"url":null,"abstract":"The problem of enumerating all connected induced subgraphs of a given order $k$ from a given graph arises in many practical applications: bioinformatics, information retrieval, processor design,to name a few. The upper bound on the number of connected induced subgraphs of order $k$ is $n\\cdot\\frac{(e\\Delta)^{k}}{(\\Delta-1)k}$, where $\\Delta$ is the maximum degree in the input graph $G$ and $n$ is the number of vertices in $G$. In this short communication, we first introduce a new neighborhood operator that is the key to design reverse search algorithms for enumerating all connected induced subgraphs of order $k$. Based on the proposed neighborhood operator, three algorithms with delay of $O(k\\cdot min\\{(n-k),k\\Delta\\}\\cdot(k\\log{\\Delta}+\\log{n}))$, $O(k\\cdot min\\{(n-k),k\\Delta\\}\\cdot n)$ and $O(k^2\\cdot min\\{(n-k),k\\Delta\\}\\cdot min\\{k,\\Delta\\})$ respectively are proposed. The first two algorithms require exponential space to improve upon the current best delay bound $O(k^2\\Delta)$\\cite{4} for this problem in the case $k>\\frac{n\\log{\\Delta}-\\log{n}-\\Delta+\\sqrt{n\\log{n}\\log{\\Delta}}}{\\log{\\Delta}}$ and $k>\\frac{n^2}{n+\\Delta}$ respectively.","PeriodicalId":13545,"journal":{"name":"Inf. Process. Lett.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inf. Process. Lett.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2139/ssrn.4150167","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The problem of enumerating all connected induced subgraphs of a given order $k$ from a given graph arises in many practical applications: bioinformatics, information retrieval, processor design,to name a few. The upper bound on the number of connected induced subgraphs of order $k$ is $n\cdot\frac{(e\Delta)^{k}}{(\Delta-1)k}$, where $\Delta$ is the maximum degree in the input graph $G$ and $n$ is the number of vertices in $G$. In this short communication, we first introduce a new neighborhood operator that is the key to design reverse search algorithms for enumerating all connected induced subgraphs of order $k$. Based on the proposed neighborhood operator, three algorithms with delay of $O(k\cdot min\{(n-k),k\Delta\}\cdot(k\log{\Delta}+\log{n}))$, $O(k\cdot min\{(n-k),k\Delta\}\cdot n)$ and $O(k^2\cdot min\{(n-k),k\Delta\}\cdot min\{k,\Delta\})$ respectively are proposed. The first two algorithms require exponential space to improve upon the current best delay bound $O(k^2\Delta)$\cite{4} for this problem in the case $k>\frac{n\log{\Delta}-\log{n}-\Delta+\sqrt{n\log{n}\log{\Delta}}}{\log{\Delta}}$ and $k>\frac{n^2}{n+\Delta}$ respectively.