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DOI: 10.4230/LIPIcs.ITCS.2024.73

Dynamic Maximal Matching in Clique Networks

Authors: Minming Li, Peter Robinson, and Xianbin Zhu

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract

We consider the problem of computing a maximal matching with a distributed algorithm in the presence of batch-dynamic changes to the graph topology. We assume that a graph of n nodes is vertex-partitioned among k players that communicate via message passing. Our goal is to provide an efficient algorithm that quickly updates the matching even if an adversary determines batches of 𝓁 edge insertions or deletions. We first show a lower bound of Ω((𝓁 log k)/(k²log n)) rounds for recomputing a matching assuming an oblivious adversary who is unaware of the initial (random) vertex partition as well as the current state of the players, and a stronger lower bound of Ω(𝓁/(klog n)) rounds against an adaptive adversary, who may choose any balanced (but not necessarily random) vertex partition initially and who knows the current state of the players. We also present a randomized algorithm that has an initialization time of O(n/(k log n)) rounds, while achieving an update time that that is independent of n: In more detail, the update time is O(⌈𝓁/k⌉ log k) against an oblivious adversary, who must fix all updates in advance. If we consider the stronger adaptive adversary, the update time becomes O (⌈𝓁/√k⌉ log k) rounds.

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Minming Li, Peter Robinson, and Xianbin Zhu. Dynamic Maximal Matching in Clique Networks. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 73:1-73:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{li_et_al:LIPIcs.ITCS.2024.73,
  author =	{Li, Minming and Robinson, Peter and Zhu, Xianbin},
  title =	{{Dynamic Maximal Matching in Clique Networks}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{73:1--73:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.73},
  URN =		{urn:nbn:de:0030-drops-196017},
  doi =		{10.4230/LIPIcs.ITCS.2024.73},
  annote =	{Keywords: distributed graph algorithm, dynamic network, maximal matching, randomized algorithm, lower bound}
}

Document

DOI: 10.4230/LIPIcs.ESA.2023.78

Massively Parallel Algorithms for the Stochastic Block Model

Authors: Zelin Li, Pan Peng, and Xianbin Zhu

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract

Learning the community structure of a large-scale graph is a fundamental problem in machine learning, computer science and statistics. Among others, the Stochastic Block Model (SBM) serves a canonical model for community detection and clustering, and the Massively Parallel Computation (MPC) model is a mathematical abstraction of real-world parallel computing systems, which provides a powerful computational framework for handling large-scale datasets. We study the problem of exactly recovering the communities in a graph generated from the SBM in the MPC model. Specifically, given kn vertices that are partitioned into k equal-sized clusters (i.e., each has size n), a graph on these kn vertices is randomly generated such that each pair of vertices is connected with probability p if they are in the same cluster and with probability q if not, where p > q > 0. We give MPC algorithms for the SBM in the (very general) s-space MPC model, where each machine is guaranteed to have memory s = Ω(log n). Under the condition that (p-q)/√p ≥ Ω̃(k^{1/2} n^{-1/2+1/(2(r-1))}) for any integer r ∈ [3,O(log n)], our first algorithm exactly recovers all the k clusters in O(kr log_s n) rounds using Õ(m) total space, or in O(rlog_s n) rounds using Õ(km) total space. If (p-q)/√p ≥ Ω̃(k^{3/4} n^{-1/4}), our second algorithm achieves O(log_s n) rounds and Õ(m) total space complexity. Both algorithms significantly improve upon a recent result of Cohen-Addad et al. [PODC'22], who gave algorithms that only work in the sublinear space MPC model, where each machine has local memory s = O(n^δ) for some constant δ > 0, with a much stronger condition on p,q,k. Our algorithms are based on collecting the r-step neighborhood of each vertex and comparing the difference of some statistical information generated from the local neighborhoods for each pair of vertices. To implement the clustering algorithms in parallel, we present efficient approaches for implementing some basic graph operations in the s-space MPC model.

Cite as

Zelin Li, Pan Peng, and Xianbin Zhu. Massively Parallel Algorithms for the Stochastic Block Model. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 78:1-78:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{li_et_al:LIPIcs.ESA.2023.78,
  author =	{Li, Zelin and Peng, Pan and Zhu, Xianbin},
  title =	{{Massively Parallel Algorithms for the Stochastic Block Model}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{78:1--78:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.78},
  URN =		{urn:nbn:de:0030-drops-187313},
  doi =		{10.4230/LIPIcs.ESA.2023.78},
  annote =	{Keywords: Massively Parallel Computation, Stochastic Block Model, Graph Algorithms}
}

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Documents authored by Zhu, Xianbin

Dynamic Maximal Matching in Clique Networks

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Massively Parallel Algorithms for the Stochastic Block Model

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