  The maximum coverage problem is to select $k$ sets from a collection of sets
such that the cardinality of the union of the selected sets is maximized. We
consider $(1-1/e-\epsilon)$-approximation algorithms for this NP-hard problem
in three standard data stream models.
  1. {\em Dynamic Model.} The stream consists of a sequence of sets being
inserted and deleted. Our multi-pass algorithm uses $\epsilon^{-2} k \cdot
\text{polylog}(n,m)$ space. The best previous result (Assadi and Khanna, SODA
2018) used $(n +\epsilon^{-4} k) \text{polylog}(n,m)$ space. While both
algorithms use $O(\epsilon^{-1} \log n)$ passes, our analysis shows that when
$\epsilon$ is a constant, it is possible to reduce the number of passes by a
$1/\log \log n$ factor without incurring additional space.
  2. {\em Random Order Model.} In this model, there are no deletions and the
sets forming the instance are uniformly randomly permuted to form the input
stream. We show that a single pass and $k \text{polylog}(n,m)$ space suffices
for arbitrary small constant $\epsilon$. The best previous result, by Warneke
et al.~(ESA 2023), used $k^2 \text{polylog}(n,m)$ space.
  3. {\em Insert-Only Model.} Lastly, our results, along with numerous previous
results, use a sub-sampling technique introduced by McGregor and Vu (ICDT 2017)
to sparsify the input instance. We explain how this technique and others used
in the paper can be implemented such that the amortized update time of our
algorithm is polylogarithmic. This also implies an improvement of the
state-of-the-art insert only algorithms in terms of the update time:
$\text{polylog}(m,n)$ update time suffices whereas the best previous result by
Jaud et al.~(SEA 2023) required update time that was linear in $k$.
