Igor Nunes , Giulio Iacobelli , Daniel Ratton Figueiredo
{"title":"A transient equivalence between Aldous-Broder and Wilson's algorithms and a two-stage framework for generating uniform spanning trees","authors":"Igor Nunes , Giulio Iacobelli , Daniel Ratton Figueiredo","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":"348 2","pages":"Article 114285"},"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.
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.