  In the semi-streaming model for processing massive graphs, an algorithm makes
multiple passes over the edges of a given $n$-vertex graph and is tasked with
computing the solution to a problem using $O(n \cdot \text{polylog}(n))$ space.
Semi-streaming algorithms for Maximal Independent Set (MIS) that run in
$O(\log\log{n})$ passes have been known for almost a decade, however, the best
lower bounds can only rule out single-pass algorithms. We close this large gap
by proving that the current algorithms are optimal: Any semi-streaming
algorithm for finding an MIS with constant probability of success requires
$\Omega(\log\log{n})$ passes. This settles the complexity of this fundamental
problem in the semi-streaming model, and constitutes one of the first optimal
multi-pass lower bounds in this model.
  We establish our result by proving an optimal round vs communication tradeoff
for the (multi-party) communication complexity of MIS. The key ingredient of
this result is a new technique, called hierarchical embedding, for performing
round elimination: we show how to pack many but small hard $(r-1)$-round
instances of the problem into a single $r$-round instance, in a way that
enforces any $r$-round protocol to effectively solve all these $(r-1)$-round
instances also. These embeddings are obtained via a novel application of
results from extremal graph theory -- in particular dense graphs with many
disjoint unique shortest paths -- together with a newly designed graph product,
and are analyzed via information-theoretic tools such as direct-sum and message
compression arguments.
