An Optimal Algorithm for Checking Regularity

We present a deterministic algorithm ${\cal A}$ that, in O(m2) time, verifies whether a given m by m bipartite graph G is regular, in the sense of Szemeredi [Regular partitions of graphs, in Problemes Combinatoires et Theorie des Graphes (Orsay, 1976), Colloques Internationaux CNRS 260, CNRS, Paris, 1978, pp. 399--401]. In the case in which G is not regular enough, our algorithm outputs a witness to this irregularity. Algorithm ${\cal A}$ may be used as a subroutine in an algorithm that finds an $\varepsilon$-regular partition of a given n-vertex graph $\Gamma$ 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 et al. [The algorithmic aspects of the regularity lemma, J. Algorithms, 16 (1994), pp. 80--109], 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.