Random Walks on Rotating Expanders

Gil Cohen, Gal Maor

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Random walks on expanders are a powerful tool which found applications in many areas of theoretical computer science, and beyond. However, they come with an inherent cost-the spectral expansion of the corresponding power graph deteriorates at a rate that is exponential in the length of the walk. As an example, when G is a d-regular Ramanujan graph, the power graph Gt has spectral expansion 2ω(t) D, where D = dt is the regularity of Gt, thus, Gt is 2ω(t) away from being Ramanujan. This exponential blowup manifests itself in many applications. In this work we bypass this barrier by permuting the vertices of the given graph after each random step. We prove that there exists a sequence of permutations for which the spectral expansion deteriorates by only a linear factor in t. In the Ramanujan case this yields an expansion of O(t D). We stress that the permutations are tailor-made to the graph at hand and require no randomness to generate. Our proof, which holds for all sufficiently high girth graphs, makes heavy use of the powerful framework of finite free probability and interlacing families that was developed in a seminal sequence of works by Marcus, Spielman and Srivastava.

Original languageEnglish
Title of host publicationSTOC 2023 - Proceedings of the 55th Annual ACM Symposium on Theory of Computing
EditorsBarna Saha, Rocco A. Servedio
PublisherAssociation for Computing Machinery
Pages971-984
Number of pages14
ISBN (Electronic)9781450399135
DOIs
StatePublished - 2 Jun 2023
Event55th Annual ACM Symposium on Theory of Computing, STOC 2023 - Orlando, United States
Duration: 20 Jun 202323 Jun 2023

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference55th Annual ACM Symposium on Theory of Computing, STOC 2023
Country/TerritoryUnited States
CityOrlando
Period20/06/2323/06/23

Funding

FundersFunder number
ERC949499

    Keywords

    • expander graphs
    • finite free probability
    • interlacing families
    • random walks on graphs
    • spectral graph theory

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