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Practical Algorithms for Low-Discrepancy 2-Colorings

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A Panorama of Discrepancy Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2107))

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Abstract

We present practical approaches for low-discrepancy 2-colorings in the hypergraph of arithmetic progressions. A simple randomized algorithm, a deterministic combinatorial algorithm (Sárközy 1974), and three estimation of distribution algorithms are compared. The best of them experimentally achieves a constant-factor approximation.

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Notes

  1. 1.

    We have 1 TiB = 240 byte and 1 PiB = 250 byte.

  2. 2.

    The description given in [6] reads differently to what we present here, since there, APs live on \([n] =\{ 1,\mathop{\ldots },n\}\) and not \([\![n]\!] =\{ 0,\mathop{\ldots },n - 1\}\), as here. Still, the generating coloring is defined on \([\![p]\!]\) in [6].

  3. 3.

    http://www-sop.inria.fr/indes/fp/Bigloo/.

  4. 4.

    However, in [18] it is suggested to use groups with vQEA in future work.

  5. 5.

    Even more, these 2 h is only the time spent in fitness function evaluation. Total running time was about 4 h, but we suspect this to be partly due to our implementation being not particularly suited for vQEA resulting in communication overhead.

  6. 6.

    http://www.informatik.uni-kiel.de/~lki/discap-results.html.

  7. 7.

    http://michaelnielsen.org/polymath1/index.php?title=The_Erd.

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Acknowledgements

I thank the editors for inviting me to contribute this chapter to “A Panorama of Discrepancy Theory”. I thank Volkmar Sauerland for proofreading. I thank my co-authors from our SEA 2013 publication [11] for joint work. Financial support through DFG Priority Program “Algorithm Engineering” (Grant Sr7/12-3) is also gratefully acknowledged.

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Authors and Affiliations

  1. Department of Computer Science, Kiel University, Christian-Albrechts-Platz 4, 24118, Kiel, Germany

    Lasse Kliemann

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  1. Lasse Kliemann
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Correspondence to Lasse Kliemann .

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Editors and Affiliations

  1. Department of Mathematics, Macquarie University, Sydney, New South Wales, Australia

    William Chen

  2. Department of Computer Science, Kiel University, Kiel, Germany

    Anand Srivastav

  3. Dipartimento di Statistica e Metodi Quantitativi, Università di Milano-Bicocca, Milano, Italy

    Giancarlo Travaglini

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Kliemann, L. (2014). Practical Algorithms for Low-Discrepancy 2-Colorings. In: Chen, W., Srivastav, A., Travaglini, G. (eds) A Panorama of Discrepancy Theory. Lecture Notes in Mathematics, vol 2107. Springer, Cham. https://doi.org/10.1007/978-3-319-04696-9_7

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