A Sequential Algorithm for Generating Random Graphs

  • Mohsen Bayati
  • Jeong Han Kim
  • Amin Saberi
Part of the Lecture Notes in Computer Science book series (LNCS, 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(md 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(m3n2). 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(n1/2 − τ), our algorithm can generate an asymptotically uniform d-regular graph. Our results are improving the previous bound of d = O(n1/3 − τ) due to Kim and Vu [30] for regular graphs.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Mohsen Bayati
    • 1
  • Jeong Han Kim
    • 2
  • Amin Saberi
    • 1
  1. 1.Stanford UniversityUSA
  2. 2.Yonsei UniversitySouth Korea

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