  We present $O(\log\log n)$-round algorithms in the Massively Parallel
Computation (MPC) model, with $\tilde{O}(n)$ memory per machine, that compute a
maximal independent set, a $1+\epsilon$ approximation of maximum matching, and
a $2+\epsilon$ approximation of minimum vertex cover, for any $n$-vertex graph
and any constant $\epsilon>0$. These improve the state of the art as follows:
  - Our MIS algorithm leads to a simple $O(\log\log \Delta)$-round MIS
algorithm in the Congested Clique model of distributed computing, which
improves on the $\tilde{O}(\sqrt{\log \Delta})$-round algorithm of Ghaffari
[PODC'17].
  - Our $O(\log\log n)$-round $(1+\epsilon)$-approximate maximum matching
algorithm simplifies or improves on the following prior work: $O(\log^2\log
n)$-round $(1+\epsilon)$-approximation algorithm of Czumaj et al. [STOC'18] and
$O(\log\log n)$-round $(1+\epsilon)$-approximation algorithm of Assadi et al.
[SODA'19].
  - Our $O(\log\log n)$-round $(2+\epsilon)$-approximate minimum vertex cover
algorithm improves on an $O(\log\log n)$-round $O(1)$-approximation of Assadi
et al. [arXiv'17].
