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A Sequential Algorithm for Generating Random Graphs

  • Conference paper
Book cover Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2007, RANDOM 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4627))

Abstract

We present the fastest FPRAS for counting and randomly generating simple graphs with a given degree sequence in a certain range. For degree sequence \((d_i)_{i=1}^n\) with maximum degree \(d_{\max}=O(m^{1/4-\tau})\), our algorithm generates almost uniform random graph with that degree sequence in time O(m d max ) where is the number of edges in the graph and τ is any positive constant. The fastest known FPRAS for this problem [22] has running time of O(m 3 n 2). Our method also gives an independent proof of McKay’s estimate [33] for the number of such graphs.

Our approach is based on sequential importance sampling (SIS) technique that has been recently successful for counting graphs [15,11,10]. Unfortunately validity of the SIS method is only known through simulations and our work together with [10] are the first results that analyze the performance of this method.

Moreover, we show that for d = O(n 1/2 − τ), our algorithm can generate an asymptotically uniform d-regular graph. Our results are improving the previous bound of d = O(n 1/3 − τ) due to Kim and Vu [30] for regular graphs.

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Author information

Authors and Affiliations

  1. Stanford University, USA

    Mohsen Bayati & Amin Saberi

  2. Yonsei University, South Korea

    Jeong Han Kim

Authors
  1. Mohsen Bayati
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  2. Jeong Han Kim
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  3. Amin Saberi
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Editor information

Editors and Affiliations

  1. Computer Science Dept., Princeton University, 35 Olden Street, NJ 08540-5233, Princeton, USA

    Moses Charikar

  2. Institut für Informatik, Christian-Albrechts-Universität zu Kiel,, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Omer Reingold

  4. Centre Universitaire d’Informatique,, Battelle bâtiment A, route de Drize 7, 1227, Geneva, Carouge, Switzerland

    José D. P. Rolim

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© 2007 Springer-Verlag Berlin Heidelberg

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Bayati, M., Kim, J.H., Saberi, A. (2007). A Sequential Algorithm for Generating Random Graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_24

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  • DOI: https://doi.org/10.1007/978-3-540-74208-1_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74207-4

  • Online ISBN: 978-3-540-74208-1

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