{"title":"小顶点切割的近最优分布计算","authors":"Merav Parter, Asaf Petruschka","doi":"10.1007/s00446-023-00455-z","DOIUrl":null,"url":null,"abstract":"<p>We present near-optimal algorithms for detecting small vertex cuts in the <span>\\({\\textsf{CONGEST}}\\)</span> model of distributed computing. Despite extensive research in this area, our understanding of the <i>vertex</i> connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, <span>\\(\\Delta \\)</span>. Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a <i>single</i> cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing <span>\\(\\Delta \\)</span> barrier. As a warm-up to our approach, we show a simple <span>\\(\\widetilde{O}(D)\\)</span>-round randomized algorithm for computing all cut vertices in a <i>D</i>-diameter <i>n</i>-vertex graph. This improves upon the <span>\\(O(D+\\Delta /\\log n)\\)</span>-round algorithm of [Pritchard and Thurimella, ICALP 2008]. Our key technical contribution is an <span>\\(\\widetilde{O}(D)\\)</span>-round randomized algorithm for computing all cut <i>pairs</i> in the graph, improving upon the state-of-the-art <span>\\(O(\\Delta \\cdot D)^4\\)</span>-round algorithm by [Parter, DISC ’19]. Note that even for the considerably simpler setting of <i>edge</i> cuts, currently <span>\\(\\widetilde{O}(D)\\)</span>-round algorithms are known <i>only</i> for detecting pairs of cut edges. Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981]. Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of <span>\\(G {\\setminus } \\{x,y\\}\\)</span> for every pair <span>\\(x,y \\in V\\)</span>, using <span>\\(\\widetilde{O}(D)\\)</span>-rounds. We believe that the tools provided in this paper are useful for omitting the <span>\\(\\Delta \\)</span>-dependency even for larger cut values.</p>","PeriodicalId":50569,"journal":{"name":"Distributed Computing","volume":"9 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Near-optimal distributed computation of small vertex cuts\",\"authors\":\"Merav Parter, Asaf Petruschka\",\"doi\":\"10.1007/s00446-023-00455-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We present near-optimal algorithms for detecting small vertex cuts in the <span>\\\\({\\\\textsf{CONGEST}}\\\\)</span> model of distributed computing. Despite extensive research in this area, our understanding of the <i>vertex</i> connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, <span>\\\\(\\\\Delta \\\\)</span>. Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a <i>single</i> cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing <span>\\\\(\\\\Delta \\\\)</span> barrier. As a warm-up to our approach, we show a simple <span>\\\\(\\\\widetilde{O}(D)\\\\)</span>-round randomized algorithm for computing all cut vertices in a <i>D</i>-diameter <i>n</i>-vertex graph. This improves upon the <span>\\\\(O(D+\\\\Delta /\\\\log n)\\\\)</span>-round algorithm of [Pritchard and Thurimella, ICALP 2008]. Our key technical contribution is an <span>\\\\(\\\\widetilde{O}(D)\\\\)</span>-round randomized algorithm for computing all cut <i>pairs</i> in the graph, improving upon the state-of-the-art <span>\\\\(O(\\\\Delta \\\\cdot D)^4\\\\)</span>-round algorithm by [Parter, DISC ’19]. Note that even for the considerably simpler setting of <i>edge</i> cuts, currently <span>\\\\(\\\\widetilde{O}(D)\\\\)</span>-round algorithms are known <i>only</i> for detecting pairs of cut edges. Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981]. Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of <span>\\\\(G {\\\\setminus } \\\\{x,y\\\\}\\\\)</span> for every pair <span>\\\\(x,y \\\\in V\\\\)</span>, using <span>\\\\(\\\\widetilde{O}(D)\\\\)</span>-rounds. We believe that the tools provided in this paper are useful for omitting the <span>\\\\(\\\\Delta \\\\)</span>-dependency even for larger cut values.</p>\",\"PeriodicalId\":50569,\"journal\":{\"name\":\"Distributed Computing\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2023-07-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Distributed Computing\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1007/s00446-023-00455-z\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Distributed Computing","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1007/s00446-023-00455-z","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Near-optimal distributed computation of small vertex cuts
We present near-optimal algorithms for detecting small vertex cuts in the \({\textsf{CONGEST}}\) model of distributed computing. Despite extensive research in this area, our understanding of the vertex connectivity of a graph is still incomplete, especially in the distributed setting. To this date, all distributed algorithms for detecting cut vertices suffer from an inherent dependency in the maximum degree of the graph, \(\Delta \). Hence, in particular, there is no truly sub-linear time algorithm for this problem, not even for detecting a single cut vertex. We take a new algorithmic approach for vertex connectivity which allows us to bypass the existing \(\Delta \) barrier. As a warm-up to our approach, we show a simple \(\widetilde{O}(D)\)-round randomized algorithm for computing all cut vertices in a D-diameter n-vertex graph. This improves upon the \(O(D+\Delta /\log n)\)-round algorithm of [Pritchard and Thurimella, ICALP 2008]. Our key technical contribution is an \(\widetilde{O}(D)\)-round randomized algorithm for computing all cut pairs in the graph, improving upon the state-of-the-art \(O(\Delta \cdot D)^4\)-round algorithm by [Parter, DISC ’19]. Note that even for the considerably simpler setting of edge cuts, currently \(\widetilde{O}(D)\)-round algorithms are known only for detecting pairs of cut edges. Our approach is based on employing the well-known linear graph sketching technique [Ahn, Guha and McGregor, SODA 2012] along with the heavy-light tree decomposition of [Sleator and Tarjan, STOC 1981]. Combining this with a careful characterization of the survivable subgraphs, allows us to determine the connectivity of \(G {\setminus } \{x,y\}\) for every pair \(x,y \in V\), using \(\widetilde{O}(D)\)-rounds. We believe that the tools provided in this paper are useful for omitting the \(\Delta \)-dependency even for larger cut values.
期刊介绍:
The international journal Distributed Computing provides a forum for original and significant contributions to the theory, design, specification and implementation of distributed systems.
Topics covered by the journal include but are not limited to:
design and analysis of distributed algorithms;
multiprocessor and multi-core architectures and algorithms;
synchronization protocols and concurrent programming;
distributed operating systems and middleware;
fault-tolerance, reliability and availability;
architectures and protocols for communication networks and peer-to-peer systems;
security in distributed computing, cryptographic protocols;
mobile, sensor, and ad hoc networks;
internet applications;
concurrency theory;
specification, semantics, verification, and testing of distributed systems.
In general, only original papers will be considered. By virtue of submitting a manuscript to the journal, the authors attest that it has not been published or submitted simultaneously for publication elsewhere. However, papers previously presented in conference proceedings may be submitted in enhanced form. If a paper has appeared previously, in any form, the authors must clearly indicate this and provide an account of the differences between the previously appeared form and the submission.