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Spectral Sparsification via Bounded-Independence Sampling

Authors Dean Doron, Jack Murtagh, Salil Vadhan, David Zuckerman

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Spectral sparsification
  • Derandomization
  • Space complexity

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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.

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Dean Doron, Jack Murtagh, Salil Vadhan, and David Zuckerman. Spectral Sparsification via Bounded-Independence Sampling. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.39

Author Details

Dean Doron
  • Department of Computer Science, Stanford University, CA, USA
Jack Murtagh
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Salil Vadhan
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
David Zuckerman
  • Department of Computer Science, University of Texas at Austin, TX, USA

Funding

  • Doron, Dean: Research supported by a Motwani Postdoctoral Fellowship.
  • Murtagh, Jack: Research supported by NSF grant CCF-1763299.
  • Vadhan, Salil: Research supported by NSF grant CCF-1763299 and a Simons Investigator Award.
  • Zuckerman, David: Research supported in part by NSF Grant CCF-1705028 and a Simons Investigator Award (#409864).

Acknowledgements

We thank Jelani Nelson for his insights on spectral sparsification via k-wise independent sampling. We also thank Jarosław Błasiok for helpful discussions about random matrices. The first author would like to thank Tselil Schramm and Amnon Ta-Shma for interesting conversations.

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