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Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs

Authors Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, Huacheng Yu

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Author Details

Lijie Chen
  • MIT, Cambridge, MA, USA
Gillat Kol
  • Princeton University, NJ, USA
Dmitry Paramonov
  • Princeton University, NJ, USA
Raghuvansh R. Saxena
  • Princeton University, NJ, USA
Zhao Song
  • Institute for Advanced Study, Princeton, NJ, US
Huacheng Yu
  • Princeton University, NJ, USA

Funding

  • Chen, Lijie: Lijie Chen is supported by an IBM Fellowship.
  • Song, Zhao: Zhao Song is supported in part by Schmidt Foundation, Simons Foundation, NSF, DARPA/SRC, Google and Amazon AWS.

Acknowledgements

We would like to thank Rajesh Jayaram for discussions on 𝓁₁ heavy hitters.

Cite As Get BibTex

Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu. Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 52:1-52:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ICALP.2021.52

Abstract

For a directed graph G with n vertices and a start vertex u_start, we wish to (approximately) sample an L-step random walk over G starting from u_start with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be Θ̃(n ⋅ L). Prior to our work, a better space complexity was only known with Õ(√L) passes. 
We essentially settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is Θ̃(n ⋅ √L), by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes p, and shows that any p-pass algorithm for this problem uses Ω̃(n ⋅ L^{1/p}) space. In addition, we show a similar Θ̃(n ⋅ √L) bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an L-step random walk from every vertex in the graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • streaming algorithms
  • random walk sampling

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