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Expander Decomposition in Dynamic Streams

Authors: Arnold Filtser, Michael Kapralov, and Mikhail Makarov

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)


Abstract
In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning 𝒞 of vertices V such that the subgraphs of G induced by the clusters C ∈ 𝒞 are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (δ, ε)-multiplicative/additive error with high probability. The power cut sparsifier uses Õ(n/εδ) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant δ.

Cite as

Arnold Filtser, Michael Kapralov, and Mikhail Makarov. Expander Decomposition in Dynamic Streams. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 50:1-50:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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@InProceedings{filtser_et_al:LIPIcs.ITCS.2023.50,
  author =	{Filtser, Arnold and Kapralov, Michael and Makarov, Mikhail},
  title =	{{Expander Decomposition in Dynamic Streams}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{50:1--50:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.50},
  URN =		{urn:nbn:de:0030-drops-175534},
  doi =		{10.4230/LIPIcs.ITCS.2023.50},
  annote =	{Keywords: Streaming, expander decomposition, graph sparsifiers}
}
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