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DOI: 10.4230/LIPIcs.ICALP.2020.39
URN: urn:nbn:de:0030-drops-124462
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12446/
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Doron, Dean ; Murtagh, Jack ; Vadhan, Salil ; Zuckerman, David

Spectral Sparsification via Bounded-Independence Sampling

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LIPIcs-ICALP-2020-39.pdf (0.6 MB)

Abstract

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph G on n vertices described by a binary string of length N, an integer k ≤ log n and an error parameter ε > 0, our algorithm runs in space Õ(k log(N w_max/w_min)) where w_max and w_min are the maximum and minimum edge weights in G, and produces a weighted graph H with Õ(n^(1+2/k)/ε²) edges that spectrally approximates G, in the sense of Spielmen and Teng [Spielman and Teng, 2004], up to an error of ε.
Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava’s effective resistance based edge sampling algorithm [Spielman and Srivastava, 2011] and uses results from recent work on space-bounded Laplacian solvers [Jack Murtagh et al., 2017]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by k above, and the resulting sparsity that can be achieved.

BibTeX - Entry

@InProceedings{doron_et_al:LIPIcs:2020:12446,
  author =	{Dean Doron and Jack Murtagh and Salil Vadhan and David Zuckerman},
  title =	{{Spectral Sparsification via Bounded-Independence Sampling}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{39:1--39:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12446},
  URN =		{urn:nbn:de:0030-drops-124462},
  doi =		{10.4230/LIPIcs.ICALP.2020.39},
  annote =	{Keywords: Spectral sparsification, Derandomization, Space complexity}
}

Keywords: Spectral sparsification, Derandomization, Space complexity
Collection: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue Date: 2020
Date of publication: 29.06.2020

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