在蒙日有向无环图中寻找最短 $M$ 链接路径

arXiv - CS - Data Structures and Algorithms Pub Date : 2024-08-01 DOI:arxiv-2408.00227

Joy Z. Wan

{"title":"在蒙日有向无环图中寻找最短 $M$ 链接路径","authors":"Joy Z. Wan","doi":"arxiv-2408.00227","DOIUrl":null,"url":null,"abstract":"A Monge directed acyclic graph (DAG) $G$ on the nodes $1,2,\\cdots,N$ has\nedges $\\left( i,j\\right) $ for $1\\leq i<j\\leq N$ carrying submodular\nedge-lengths. Finding a shortest $M$-link path from $1$ to $N$ in $G$ for any\ngiven $1<M<N-1$ has many applications. In this paper, we give a\ncontract-and-conquer algorithm for this problem which runs in $O\\left(\n\\sqrt{NM\\left( N-M\\right) \\log\\left( N-M\\right) }\\right) $ time and $O\\left(\nN\\right) $ space. It is the first $o\\left( NM\\right) $-time algorithm with\nlinear space complexity, and its time complexity decreases with $M$ when $M\\geq\nN/2$. In contrast, all previous strongly polynomial algorithms have running\ntime growing with $M$. For both $O\\left( poly\\left( \\log N\\right) \\right) $ and\n$N-O\\left( poly\\left( \\log N\\right) \\right) $ regimes of $M$, our algorithm has\nrunning time $O\\left( N\\cdot poly\\left( \\log N\\right) \\right) $, which\npartially answers an open question rased in \\cite{AST94} affirmatively.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"4 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Finding a Shortest $M$-link Path in a Monge Directed Acyclic Graph\",\"authors\":\"Joy Z. Wan\",\"doi\":\"arxiv-2408.00227\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A Monge directed acyclic graph (DAG) $G$ on the nodes $1,2,\\\\cdots,N$ has\\nedges $\\\\left( i,j\\\\right) $ for $1\\\\leq i<j\\\\leq N$ carrying submodular\\nedge-lengths. Finding a shortest $M$-link path from $1$ to $N$ in $G$ for any\\ngiven $1<M<N-1$ has many applications. In this paper, we give a\\ncontract-and-conquer algorithm for this problem which runs in $O\\\\left(\\n\\\\sqrt{NM\\\\left( N-M\\\\right) \\\\log\\\\left( N-M\\\\right) }\\\\right) $ time and $O\\\\left(\\nN\\\\right) $ space. It is the first $o\\\\left( NM\\\\right) $-time algorithm with\\nlinear space complexity, and its time complexity decreases with $M$ when $M\\\\geq\\nN/2$. In contrast, all previous strongly polynomial algorithms have running\\ntime growing with $M$. For both $O\\\\left( poly\\\\left( \\\\log N\\\\right) \\\\right) $ and\\n$N-O\\\\left( poly\\\\left( \\\\log N\\\\right) \\\\right) $ regimes of $M$, our algorithm has\\nrunning time $O\\\\left( N\\\\cdot poly\\\\left( \\\\log N\\\\right) \\\\right) $, which\\npartially answers an open question rased in \\\\cite{AST94} affirmatively.\",\"PeriodicalId\":501525,\"journal\":{\"name\":\"arXiv - CS - Data Structures and Algorithms\",\"volume\":\"4 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Data Structures and Algorithms\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2408.00227\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Data Structures and Algorithms","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.00227","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}