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Bounds for Semi-disjoint Bilinear Forms in a Unit-Cost Computational Model

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Theory and Applications of Models of Computation (TAMC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10185))

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Abstract

We study the complexity of the so called semi-disjoint bilinear forms over different semi-rings, in particular the n-dimensional vector convolution and \(n\times n\) matrix product. We consider a powerful unit-cost computational model over the ring of integers allowing for several additional operations and generation of large integers. We show the following dichotomy for such a powerful model: while almost all arithmetic semi-disjoint bilinear forms have the same asymptotic time complexity as that yielded by naive algorithms, matrix multiplication, the so called distance matrix product, and vector convolution can be solved in a linear number of steps. It follows in particular that in order to obtain a non-trivial lower bounds for these three basic problems one has to assume restrictions on the set of allowed operations and/or the size of used integers.

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Acknowledgments

The authors are grateful to Christos Levcopoulos for valuable comments. This research has been supported in part by Swedish Research Council grant 621-2011-6179.

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Authors and Affiliations

  1. Department of Computer Science, Lund University, Lund, Sweden

    Andrzej Lingas

  2. Department of Computer Science, Malmö University, Malmö, Sweden

    Mia Persson

  3. Centre for Mathematical Sciences, Lund University, Lund, Sweden

    Dzmitry Sledneu

Authors
  1. Andrzej Lingas
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  2. Mia Persson
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  3. Dzmitry Sledneu
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Correspondence to Andrzej Lingas .

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Editors and Affiliations

  1. Anna University, Chennai, India

    T.V. Gopal

  2. Universität Bern , Bern, Switzerland

    Gerhard Jäger

  3. Universität Bern , Bern, Switzerland

    Silvia Steila

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Lingas, A., Persson, M., Sledneu, D. (2017). Bounds for Semi-disjoint Bilinear Forms in a Unit-Cost Computational Model. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_30

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  • DOI: https://doi.org/10.1007/978-3-319-55911-7_30

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-55910-0

  • Online ISBN: 978-3-319-55911-7

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