An optimal algorithm for checking regularity: (extended abstract)

  • Authors:

  • Y. Kohayakawa;V. Rödl;L. Thoma

  • Affiliations:

  • Universidade de Sāo Paulo, Rua do Matāo 1010, 05508-900 Sāo Paulo, Brazil;Emory University, Atlanta, GA;University of Rhode Island, Kingston RI

  • Venue:
  • SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2002

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Abstract

We present a deterministic algorithm A that, in O(m2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemerédi [18]. In the case in which G is not regular enough, our algorithm outputs a witness to this irregularity. Algorithm A may be used as a subroutine in an algorithm that finds an ε-regular partition of a given n-vertex graph Γ in time O(n2). This time complexity is optimal, up to a constant factor, and improves upon the bound O(M(n)), proved by Alon, Duke, Lefmann, Rödl, and Yuster [1, 2], where M(n) = O(n2.376) is the time required to square a 0-1 matrix over the integers.Our approach is elementary, except that it makes use of linear-sized expanders to accomplish a suitable form of deterministic sampling.