#### Title

An optimal algorithm for checking regularity

#### Document Type

Article

#### Date of Original Version

8-1-2003

#### 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 [Regular partitions of graphs, in Problèmes Combinatoires et Théorie 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 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 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.

#### Publication Title

SIAM Journal on Computing

#### Volume

32

#### Issue

5

#### Citation/Publisher Attribution

Kohayakawa, Y., V. Rödl, and L. Thoma.
"An optimal algorithm for checking regularity."
*SIAM Journal on Computing*
32,
5
(2003): 1210-1235.
doi:10.1137/S0097539702408223.