  We present a constant-round algorithm in the massively parallel computation
(MPC) model for evaluating a natural join where every input relation has two
attributes. Our algorithm achieves a load of $\tilde{O}(m/p^{1/\rho})$ where
$m$ is the total size of the input relations, $p$ is the number of machines,
$\rho$ is the join's fractional edge covering number, and $\tilde{O}(.)$ hides
a polylogarithmic factor. The load matches a known lower bound up to a
polylogarithmic factor. At the core of the proposed algorithm is a new theorem
(which we name the "isolated cartesian product theorem") that provides fresh
insight into the problem's mathematical structure. Our result implies that the
subgraph enumeration problem, where the goal is to report all the occurrences
of a constant-sized subgraph pattern, can be settled optimally (up to a
polylogarithmic factor) in the MPC model.
