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DOI: 10.4230/LIPIcs.DISC.2024.12

Faster Cycle Detection in the Congested Clique

Authors: Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 319, 38th International Symposium on Distributed Computing (DISC 2024)

Abstract

We provide a fast distributed algorithm for detecting h-cycles in the Congested Clique model, whose running time decreases as the number of h-cycles in the graph increases. In undirected graphs, constant-round algorithms are known for cycles of even length. Our algorithm greatly improves upon the state of the art for odd values of h. Moreover, our running time applies also to directed graphs, in which case the improvement is for all values of h. Further, our techniques allow us to obtain a triangle detection algorithm in the quantum variant of this model, which is faster than prior work. A key technical contribution we develop to obtain our fast cycle detection algorithm is a new algorithm for computing the product of many pairs of small matrices in parallel, which may be of independent interest.

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Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams. Faster Cycle Detection in the Congested Clique. In 38th International Symposium on Distributed Computing (DISC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 319, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{censorhillel_et_al:LIPIcs.DISC.2024.12,
  author =	{Censor-Hillel, Keren and Even, Tomer and Vassilevska Williams, Virginia},
  title =	{{Faster Cycle Detection in the Congested Clique}},
  booktitle =	{38th International Symposium on Distributed Computing (DISC 2024)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-352-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{319},
  editor =	{Alistarh, Dan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2024.12},
  URN =		{urn:nbn:de:0030-drops-212382},
  doi =		{10.4230/LIPIcs.DISC.2024.12},
  annote =	{Keywords: triangle detection, cycle detection, distributed computing, Congested Clique, quantum computing, Fast matrix multiplication, Fast rectangular matrix multiplication}
}

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Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2024.37

Fast Approximate Counting of Cycles

Authors: Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal.

Cite as

Keren Censor-Hillel, Tomer Even, and Virginia Vassilevska Williams. Fast Approximate Counting of Cycles. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 37:1-37:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{censorhillel_et_al:LIPIcs.ICALP.2024.37,
  author =	{Censor-Hillel, Keren and Even, Tomer and Vassilevska Williams, Virginia},
  title =	{{Fast Approximate Counting of Cycles}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{37:1--37:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.37},
  URN =		{urn:nbn:de:0030-drops-201809},
  doi =		{10.4230/LIPIcs.ICALP.2024.37},
  annote =	{Keywords: Approximate triangle counting, Approximate cycle counting Fast matrix multiplication, Fast rectangular matrix multiplication}
}

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Faster Cycle Detection in the Congested Clique

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Fast Approximate Counting of Cycles

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