A Sequential Algorithm for Generating Random Graphs

  • Mohsen Bayati
  • Jeong Han Kim
  • Amin Saberi
Conference paper

DOI: 10.1007/978-3-540-74208-1_24

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4627)
Cite this paper as:
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. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg

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 Open image in new window 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 University 
  2. 2.Yonsei University 

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