  Maximizing a non-negative, monontone, submodular function $f$ over $n$
elements under a cardinality constraint $k$ (SMCC) is a well-studied NP-hard
problem. It has important applications in, e.g., machine learning and influence
maximization. Though the theoretical problem admits polynomial-time
approximation algorithms, solving it in practice often involves frequently
querying submodular functions that are expensive to compute. This has motivated
significant research into designing parallel approximation algorithms in the
adaptive complexity model; adaptive complexity (adaptivity) measures the number
of sequential rounds of $\text{poly}(n)$ function queries an algorithm
requires. The state-of-the-art algorithms can achieve
$(1-\frac{1}{e}-\varepsilon)$-approximate solutions with
$O(\frac{1}{\varepsilon^2}\log n)$ adaptivity, which approaches the known
adaptivity lower-bounds. However, the $O(\frac{1}{\varepsilon^2} \log n)$
adaptivity only applies to maximizing worst-case functions that are unlikely to
appear in practice. Thus, in this paper, we consider the special class of
$p$-superseparable submodular functions, which places a reasonable constraint
on $f$, based on the parameter $p$, and is more amenable to maximization, while
also having real-world applicability. Our main contribution is the algorithm
LS+GS, a finer-grained version of the existing LS+PGB algorithm, designed for
instances of SMCC when $f$ is $p$-superseparable; it achieves an expected
$(1-\frac{1}{e}-\varepsilon)$-approximate solution with
$O(\frac{1}{\varepsilon^2}\log(p k))$ adaptivity independent of $n$.
Additionally, unrelated to $p$-superseparability, our LS+GS algorithm uses only
$O(\frac{n}{\varepsilon} + \frac{\log n}{\varepsilon^2})$ oracle queries, which
has an improved dependence on $\varepsilon^{-1}$ over the state-of-the-art
LS+PGB; this is achieved through the design of a novel thresholding subroutine.
