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DOI: 10.4230/LIPIcs.ITCS.2026.55

Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs

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

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

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.

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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)

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@InProceedings{efremenko_et_al:LIPIcs.ITCS.2026.55,
  author =	{Efremenko, Klim and Kol, Gillat and Saxena, Raghuvansh R. and Zhang, Zhijun},
  title =	{{Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{55:1--55:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.55},
  URN =		{urn:nbn:de:0030-drops-253423},
  doi =		{10.4230/LIPIcs.ITCS.2026.55},
  annote =	{Keywords: Random Walk, streaming Algorithm, universal Optimality}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2025.22

Round-Vs-Resilience Tradeoffs for Binary Feedback Channels

Authors: Mark Braverman, Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract

In a celebrated result from the 60’s, Berlekamp showed that feedback can be used to increase the maximum fraction of adversarial noise that can be tolerated by binary error correcting codes from 1/4 to 1/3. However, his result relies on the assumption that feedback is "continuous", i.e., after every utilization of the channel, the sender gets the symbol received by the receiver. While this assumption is natural in some settings, in other settings it may be unreasonable or too costly to maintain. In this work, we initiate the study of round-restricted feedback channels, where the number r of feedback rounds is possibly much smaller than the number of utilizations of the channel. Error correcting codes for such channels are protocols where the sender can ask for feedback at most r times, and, upon a feedback request, it obtains all the symbols received since its last feedback request. We design such error correcting protocols for both the adversarial binary erasure channel and for the adversarial binary corruption (bit flip) channel. For the erasure channel, we give an exact characterization of the round-vs-resilience tradeoff by designing a (constant rate) protocol with r feedback rounds, for every r, and proving that its noise resilience is optimal. Designing such error correcting protocols for the corruption channel is substantially more involved. We show that obtaining the optimal resilience, even with one feedback round (r = 1), requires settling (proving or disproving) a new, seemingly unrelated, "clean" combinatorial conjecture, about the maximum cut in weighted graphs versus the "imbalance" of an average cut. Specifically, we prove an upper bound on the optimal resilience (impossibility result), and show that the existence of a matching lower bound (a protocol) is equivalent to the correctness of our conjecture.

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Mark Braverman, Klim Efremenko, Gillat Kol, Raghuvansh R. Saxena, and Zhijun Zhang. Round-Vs-Resilience Tradeoffs for Binary Feedback Channels. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 22:1-22:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{braverman_et_al:LIPIcs.ITCS.2025.22,
  author =	{Braverman, Mark and Efremenko, Klim and Kol, Gillat and Saxena, Raghuvansh R. and Zhang, Zhijun},
  title =	{{Round-Vs-Resilience Tradeoffs for Binary Feedback Channels}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{22:1--22:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.22},
  URN =		{urn:nbn:de:0030-drops-226506},
  doi =		{10.4230/LIPIcs.ITCS.2025.22},
  annote =	{Keywords: Round-restricted feedback channel, error correcting code, noise resilience}
}

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Documents authored by Zhang, Zhijun

Universally Optimal Streaming Algorithm for Random Walks in Dense Graphs

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Round-Vs-Resilience Tradeoffs for Binary Feedback Channels

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