  For the problem of maximizing a monotone, submodular function with respect to
a cardinality constraint $k$ on a ground set of size $n$, we provide an
algorithm that achieves the state-of-the-art in both its empirical performance
and its theoretical properties, in terms of adaptive complexity, query
complexity, and approximation ratio; that is, it obtains, with high
probability, query complexity of $O(n)$ in expectation, adaptivity of
$O(\log(n))$, and approximation ratio of nearly $1-1/e$. The main algorithm is
assembled from two components which may be of independent interest. The first
component of our algorithm, LINEARSEQ, is useful as a preprocessing algorithm
to improve the query complexity of many algorithms. Moreover, a variant of
LINEARSEQ is shown to have adaptive complexity of $O( \log (n / k) )$ which is
smaller than that of any previous algorithm in the literature. The second
component is a parallelizable thresholding procedure THRESHOLDSEQ for adding
elements with gain above a constant threshold. Finally, we demonstrate that our
main algorithm empirically outperforms, in terms of runtime, adaptive rounds,
total queries, and objective values, the previous state-of-the-art algorithm
FAST in a comprehensive evaluation with six submodular objective functions.
