  Identifying clusters of similar elements in a set is a common objective in
data analysis. With the immense growth of data and physical limitations on
single processor speed, it is necessary to find efficient parallel algorithms
for clustering tasks. In this paper, we study the problem of correlation
clustering in bounded arboricity graphs with respect to the Massively Parallel
Computation (MPC) model. More specifically, we are given a complete graph where
the vertices correspond to the elements and each edge is either positive or
negative, indicating whether pairs of vertices are similar or dissimilar. The
task is to partition the vertices into clusters with as few disagreements as
possible. That is, we want to minimize the number of positive inter-cluster
edges and negative intra-cluster edges.
  Consider an input graph $G$ on $n$ vertices such that the positive edges
induce a $\lambda$-arboric graph. Our main result is a 3-approximation
(\emph{in expectation}) algorithm that runs in $\mathcal{O}(\log \lambda \cdot
\log \log n)$ MPC rounds in the \emph{sublinear memory regime}. This is
obtained by combining structural properties of correlation clustering on
bounded arboricity graphs with the insights of Fischer and Noever (SODA '18) on
randomized greedy MIS and the \PIVOT algorithm of Ailon, Charikar, and Newman
(STOC '05). Combined with known graph matching algorithms, our structural
property also implies an exact algorithm and algorithms with \emph{worst case}
$(1+\epsilon)$-approximation guarantees in the special case of forests, where
$\lambda=1$.
