小顶点切割的近最优分布计算

IF 1.3 4区 计算机科学 Q3 COMPUTER SCIENCE, THEORY & METHODS Distributed Computing Pub Date : 2023-07-14 DOI:10.1007/s00446-023-00455-z
Merav Parter, Asaf Petruschka
{"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}
引用次数: 0

摘要

我们提出了在分布式计算\({\textsf{CONGEST}}\)模型中检测小顶点切割的近最优算法。尽管在这一领域进行了广泛的研究,但我们对图的顶点连通性的理解仍然不完整,特别是在分布式设置中。到目前为止,所有用于检测切割顶点的分布式算法都在图的最大程度上存在固有的依赖性,\(\Delta \)。因此,特别地,对于这个问题,没有真正的亚线性时间算法,甚至没有检测单个切割顶点的算法。我们采用了一种新的顶点连接算法,它允许我们绕过现有的\(\Delta \)障碍。作为我们方法的预热,我们展示了一个简单的\(\widetilde{O}(D)\) -round随机化算法,用于计算d -直径n-顶点图中的所有切割顶点。这改进了[Pritchard and Thurimella, ICALP 2008]的\(O(D+\Delta /\log n)\) -round算法。我们的关键技术贡献是一个\(\widetilde{O}(D)\) -round随机算法,用于计算图中的所有切对,改进了[partner, DISC ' 19]的最先进的\(O(\Delta \cdot D)^4\) -round算法。请注意,即使对于相当简单的边缘切割设置,目前\(\widetilde{O}(D)\) -round算法仅用于检测切割边缘对。我们的方法基于采用著名的线性图形素描技术[Ahn, Guha和McGregor, SODA 2012]以及[Sleator和Tarjan, STOC 1981]的重-轻树分解。将此与可存活子图的仔细表征相结合,使我们能够使用\(\widetilde{O}(D)\) -round确定每个对\(x,y \in V\)的\(G {\setminus } \{x,y\}\)的连通性。我们相信本文中提供的工具对于省略\(\Delta \) -依赖非常有用,即使对于较大的切割值也是如此。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

摘要图片

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
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.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Distributed Computing
Distributed Computing 工程技术-计算机:理论方法
CiteScore
3.20
自引率
0.00%
发文量
24
审稿时长
>12 weeks
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.
期刊最新文献
A wait-free queue with polylogarithmic step complexity Deterministic near-optimal distributed listing of cliques On implementing SWMR registers from SWSR registers in systems with Byzantine failures Developing Irelands first National Student Mental Health and Suicide Prevention Framework for Higher Education. Asymmetric distributed trust
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1