  Maximal Independent Set (MIS) is one of the fundamental and most well-studied
problems in distributed graph algorithms. Even after four decades of intensive
research, the best-known (randomized) MIS algorithms have $O(\log{n})$ round
complexity on general graphs [Luby, STOC 1986] (where $n$ is the number of
nodes), while the best-known lower bound is
$\Omega(\sqrt{\log{n}/\log\log{n}})$ [Kuhn, Moscibroda, Wattenhofer, JACM
2016]. Breaking past the $O(\log{n})$ round complexity upper bound or showing
stronger lower bounds have been longstanding open problems.
  Our main contribution is to show that MIS can be computed in awake complexity
that is exponentially better compared to the best known round complexity of
$O(\log n)$ and also bypassing its fundamental
$\Omega(\sqrt{\log{n}/\log\log{n}})$ round complexity lower bound
exponentially. Specifically, we show that MIS can be computed by a randomized
distributed (Monte Carlo) algorithm in $O(\log\log{n} )$ awake complexity with
high probability (i.e., with probability at least $1 - n^{-1}$). This algorithm
has a round complexity of $O((\log^7 n) \log \log n)$. We also show that we can
improve the round complexity at the cost of a slight increase in awake
complexity, by presenting a randomized distributed (Monte Carlo) algorithm for
MIS that, with high probability, computes an MIS in $O((\log\log{n})\log^*n)$
awake complexity and $O((\log^3 n) (\log \log n) \log^*n)$ round complexity.
Our algorithms work in the CONGEST model where messages of size $O(\log n)$
bits can be sent per edge per round.
