Local Access to Random Walks

Amartya Shankha Biswas, Edward Pyne, Ronitt Rubinfeld

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

Abstract

For a graph G on n vertices, naively sampling the position of a random walk of at time t requires work Ω(t). We desire local access algorithms supporting positionG(t) queries, which return the position of a random walk from some fixed start vertex s at time t, where the joint distribution of returned positions is 1/poly(n) close to those of a uniformly random walk in ℓ1 distance. We first give an algorithm for local access to random walks on a given undirected d-regular graph with (Equation presented) runtime per query, where λ is the second-largest eigenvalue of the random walk matrix of the graph in absolute value. Since random d-regular graphs G(n,d) are expanders with high probability, this gives an Õ(√n) algorithm for a graph drawn from G(n,d) whp, which improves on the naive method for small numbers of queries. We then prove that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω(√n/log(n)) per query in expectation when the input graph is drawn from G(n,d), obtaining a nearly matching lower bound. We further show an Ω(n1/4) runtime per query lower bound even with an oblivious adversary (i.e. when the query sequence is fixed in advance). We then show that for families of graphs with additional group theoretic structure, dramatically better results can be achieved. We give local access to walks on small-degree abelian Cayley graphs, including cycles and hypercubes, with runtime polylog(n) per query. This also allows for efficient local access to walks on polylog degree expanders. We show that our techniques apply to graphs with high degree by extending or results to graphs constructed using the tensor product (giving fast local access to walks on degree nϵ graphs for any ϵ ∈ (0,1]) and Cartesian product.

Original languageEnglish
Title of host publication13th Innovations in Theoretical Computer Science Conference, ITCS 2022
EditorsMark Braverman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772174
DOIs
StatePublished - 1 Jan 2022
Externally publishedYes
Event13th Innovations in Theoretical Computer Science Conference, ITCS 2022 - Berkeley, United States
Duration: 31 Jan 20223 Feb 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume215
ISSN (Print)1868-8969

Conference

Conference13th Innovations in Theoretical Computer Science Conference, ITCS 2022
Country/TerritoryUnited States
CityBerkeley
Period31/01/223/02/22

Keywords

  • Local computation
  • Random generation
  • Sublinear time algorithms

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