  We consider the problem of computing a maximal matching with a distributed
algorithm in the presence of batch-dynamic changes to the graph topology. We
assume that a graph of $n$ nodes is vertex-partitioned among $k$ players that
communicate via message passing. Our goal is to provide an efficient algorithm
that quickly updates the matching even if an adversary determines batches of
$\ell$ edge insertions or deletions.
  Assuming a link bandwidth of $O(\beta\log n)$ bits per round, for a parameter
$\beta \ge 1$, we first show a lower bound of $\Omega( \frac{\ell\,\log
k}{\beta\,k^2\log n})$ rounds for recomputing a matching assuming an oblivious
adversary who is unaware of the initial (random) vertex partition as well as
the current state of the players, and a stronger lower bound of
$\Omega(\frac{\ell}{\beta\,k\log n})$ rounds against an adaptive adversary, who
may choose any balanced (but not necessarily random) vertex partition initially
and who knows the current state of the players.
  We also present a randomized algorithm that has an initialization time of $O(
\lceil\frac{n}{\beta\,k}\rceil\log n )$ rounds, while achieving an update time
that that is independent of $n$: In more detail, the update time is $O( \lceil
\frac{\ell}{\beta\,k} \rceil \log(\beta\,k))$ against an oblivious adversary,
who must fix all updates in advance. If we consider the stronger adaptive
adversary, the update time becomes $O( \lceil
\frac{\ell}{\sqrt{\beta\,k}}\rceil \log(\beta\,k))$ rounds.
