A transient equivalence between Aldous-Broder and Wilson's algorithms and a two-stage framework for generating uniform spanning trees

IF 0.7 3区 数学 Q2 MATHEMATICS
{"title":"A transient equivalence between Aldous-Broder and Wilson's algorithms and a two-stage framework for generating uniform spanning trees","authors":"","doi":"10.1016/j.disc.2024.114285","DOIUrl":null,"url":null,"abstract":"<div><div>The <em>Aldous-Broder</em> and <em>Wilson</em> are two well-known algorithms for generating uniform spanning trees (USTs) based on random walks. This work studies their transient relationship by introducing the notion of <em>branches</em>—paths generated by the two algorithms on particular stopping times, in order to show that the trees built by the two algorithms when running on a complete graph are statistically equivalent on these stopping times. This leads to a hybrid algorithm that can generate USTs faster than either of the two algorithms. The idea is generalized to a two-stage framework to generate USTs on arbitrary graphs. The feasibility of the framework is shown through various examples, including some edge transitive graphs where the average running time can be 25% smaller than <em>Wilson</em> to generate USTs. Results obtained through numerical simulations of the framework on complete graphs and hypercubes illustrate the findings.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004163","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

The Aldous-Broder and Wilson are two well-known algorithms for generating uniform spanning trees (USTs) based on random walks. This work studies their transient relationship by introducing the notion of branches—paths generated by the two algorithms on particular stopping times, in order to show that the trees built by the two algorithms when running on a complete graph are statistically equivalent on these stopping times. This leads to a hybrid algorithm that can generate USTs faster than either of the two algorithms. The idea is generalized to a two-stage framework to generate USTs on arbitrary graphs. The feasibility of the framework is shown through various examples, including some edge transitive graphs where the average running time can be 25% smaller than Wilson to generate USTs. Results obtained through numerical simulations of the framework on complete graphs and hypercubes illustrate the findings.
阿尔多斯-布罗德算法和威尔逊算法之间的瞬时等价关系以及生成均匀生成树的两阶段框架
Aldous-Broder 算法和 Wilson 算法是基于随机游走生成均匀生成树(UST)的两种著名算法。这项研究通过引入这两种算法在特定停止时间生成的分支路径概念来研究它们之间的瞬时关系,从而证明这两种算法在完整图上运行时生成的树在这些停止时间上具有统计学等价性。这就产生了一种混合算法,它生成 UST 的速度比两种算法中的任何一种都快。这一想法被推广到一个两阶段框架,用于在任意图上生成 UST。该框架的可行性通过各种实例得到了证明,其中包括一些边缘传递图,在这些图上生成 UST 所需的平均运行时间比 Wilson 算法少 25%。该框架在完整图和超立方体上的数值模拟结果也说明了这些发现。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
期刊介绍: Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
×
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
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
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 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信