  In this paper we study the adaptivity of submodular maximization. Adaptivity
quantifies the number of sequential rounds that an algorithm makes when
function evaluations can be executed in parallel. Adaptivity is a fundamental
concept that is heavily studied across a variety of areas in computer science,
largely due to the need for parallelizing computation. For the canonical
problem of maximizing a monotone submodular function under a cardinality
constraint, it is well known that a simple greedy algorithm achieves a $1-1/e$
approximation and that this approximation is optimal for polynomial-time
algorithms. Somewhat surprisingly, despite extensive efforts on submodular
optimization for large-scale datasets, until very recently there was no known
algorithm that achieves a constant factor approximation for this problem whose
adaptivity is sublinear in the size of the ground set $n$.
  Recent work by Balkanski and Singer describes an algorithm that obtains an
approximation arbitrarily close to $1/3$ in $\mathcal{O}(\log n)$ adaptive
rounds and shows that no algorithm can obtain a constant factor approximation
in $\tilde{o}(\log n)$ adaptive rounds. This approach achieves an exponential
speedup in adaptivity (and parallel running time) at the expense of
approximation quality.
  In this paper we describe a novel approach that yields an algorithm whose
approximation is arbitrarily close to the optimal $1-1/e$ guarantee in
$\mathcal{O}(\log n)$ adaptive rounds. This algorithm therefore achieves an
exponential speedup in parallel running time for submodular maximization at the
expense of an arbitrarily small loss in approximation quality. This guarantee
is optimal in both approximation and adaptivity, up to lower order terms.
