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Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs

Authors Klim Efremenko , Gillat Kol , Raghuvansh R. Saxena, Zhijun Zhang

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Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Random Walk
  • streaming Algorithm
  • universal Optimality

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Abstract

Sampling a random walk is a fundamental primitive in many graph applications. In the streaming model, it is known that sampling an L-step random walk on an n-vertex directed graph requires Ω(n L) space, implying that no sublinear-space streaming algorithm exists for general graphs.
We show that sublinear algorithms are possible for the case of dense graphs, where every vertex has out-degree at least Ω(n). In particular, we give a one-pass turnstile streaming algorithm that uses only 𝒪̃(L) memory for such graphs. More broadly, for graphs with minimum out-degree at least d, our streaming algorithm samples a random walk using 𝒪̃(n/d ⋅ L) memory.
We show that our algorithm is optimal in a strong "beyond worst-case" sense. To formalize this, we introduce the notion of universal optimality for graph streaming algorithms. Informally, a streaming algorithm is universally optimal if it performs (almost) as well as possible on every graph, assuming a worst-case choice of the streaming order. This notion of universal optimality is a key conceptual contribution of our work.

Cite As Get BibTex

Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang. Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 55:1-55:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.55

Author Details

Klim Efremenko
  • Ben-Gurion University, Be'er-Sheva, Israel
Gillat Kol
  • Princeton University, NJ, US
Raghuvansh R. Saxena
  • Tata Institute of Fundamental Research, Mumbai, India
Zhijun Zhang
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria

Funding

  • Efremenko, Klim: Supported by the Israel Science Foundation (ISF) through grant No. 1456/18 and European Research Council Grant number: 949707.
  • Kol, Gillat: Supported by a National Science Foundation CAREER award CCF-1750443 and by a BSF grant No. 2018325.
  • Saxena, Raghuvansh R.: Supported by the Department of Atomic Energy, Government of India, under project no. RTI4001.
  • Zhang, Zhijun: This research was partially done while at Princeton University and was partially funded by the Ministry of Education and Science of Bulgaria (support for INSAIT, part of the Bulgarian National Roadmap for Research Infrastructure).

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