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A Matrix Hyperbolic Cosine Algorithm and Applications

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Automata, Languages, and Programming (ICALP 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7391))

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Abstract

In this paper, we generalize Spencer’s hyperbolic cosine algorithm to the matrix-valued setting. We apply the proposed algorithm to several problems by analyzing its computational efficiency under two special cases of matrices; one in which the matrices have a group structure and an other in which they have rank-one. As an application of the former case, we present a deterministic algorithm that, given the multiplication table of a finite group of size n, it constructs an expanding Cayley graph of logarithmic degree in near-optimal \(\mathcal{O}(n^2\log^3 n)\) time. For the latter case, we present a fast deterministic algorithm for spectral sparsification of positive semi-definite matrices, which implies an improved deterministic algorithm for spectral graph sparsification of dense graphs. In addition, we give an elementary connection between spectral sparsification of positive semi-definite matrices and element-wise matrix sparsification. As a consequence, we obtain improved element-wise sparsification algorithms for diagonally dominant-like matrices.

A full version of this paper can be found at http://arxiv.org/abs/1103.2793

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Toronto, Canada

    Anastasios Zouzias

Authors
  1. Anastasios Zouzias
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Centre for Discrete Mathematics and its Applications, University of Warwick, Warwick, UK

    Artur Czumaj

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Kurt Mehlhorn

  3. Computer Laboratory, University of Cambridge, UK

    Andrew Pitts

  4. ETH Zurich, Switzerland

    Roger Wattenhofer

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Zouzias, A. (2012). A Matrix Hyperbolic Cosine Algorithm and Applications. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_71

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  • DOI: https://doi.org/10.1007/978-3-642-31594-7_71

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31593-0

  • Online ISBN: 978-3-642-31594-7

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