An optimal algorithm for checking regularity (extended abstract)
Date of Original Version
We present a deterministic algorithm A that, in 0(m2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemeredi . In the case in which G is not regular enough, our algorithm outputs a vntness to this irregularity. Algorithm A may be used as a subroutine in an algorithm that finds an e-regular partition of a given n-vertex graph Λ in time 0(n2). This time complexity is optimal, up to a constant factor, and improves upon the bound 0(M(n)), proved by Alon, Duke, Lefmann, Rodl, and Yuster [1, 2], where M(n) = 0(n2-376) is the time required to square a 6-1 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.
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Kohayakawa, Y., V. Rodl, and L. Thoma. "An optimal algorithm for checking regularity (extended abstract)." Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms 06-08-January-2002, (2002): 277-286. https://digitalcommons.uri.edu/math_facpubs/251