{"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}
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Abstract
A Monge directed acyclic graph (DAG) $G$ on the nodes $1,2,\cdots,N$ has
edges $\left( i,j\right) $ for $1\leq i
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