  We present a new approach for solving (minimum disagreement) correlation
clustering that results in sublinear algorithms with highly efficient time and
space complexity for this problem. In particular, we obtain the following
algorithms for $n$-vertex $(+/-)$-labeled graphs $G$:
  -- A sublinear-time algorithm that with high probability returns a constant
approximation clustering of $G$ in $O(n\log^2{n})$ time assuming access to the
adjacency list of the $(+)$-labeled edges of $G$ (this is almost quadratically
faster than even reading the input once). Previously, no sublinear-time
algorithm was known for this problem with any multiplicative approximation
guarantee.
  -- A semi-streaming algorithm that with high probability returns a constant
approximation clustering of $G$ in $O(n\log{n})$ space and a single pass over
the edges of the graph $G$ (this memory is almost quadratically smaller than
input size). Previously, no single-pass algorithm with $o(n^2)$ space was known
for this problem with any approximation guarantee.
  The main ingredient of our approach is a novel connection to sparse-dense
graph decompositions that are used extensively in the graph coloring
literature. To our knowledge, this connection is the first application of these
decompositions beyond graph coloring, and in particular for the correlation
clustering problem, and can be of independent interest.
