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.

翻译：我们考虑在存在批量动态图拓扑变化的条件下，使用分布式算法计算最大匹配的问题。假设一个包含 $n$ 个节点的图为 $k$ 个通过消息传递进行通信的参与者所顶点划分。我们的目标是提供一种高效算法，即使在对手确定 $\ell$ 条边的插入或删除批次时，也能快速更新匹配。在参数 $\beta \ge 1$ 下，假设每轮链路带宽为 $O(\beta\log n)$ 比特，我们首先针对一个不知晓初始（随机）顶点划分以及参与者当前状态的遗忘对手，证明了重新计算匹配的下界为 $\Omega( \frac{\ell\,\log k}{\beta\,k^2\log n})$ 轮；针对一个可初始选择任意平衡（但不一定随机）顶点划分且知晓参与者当前状态的自适应对手，则证明了更强的下界 $\Omega(\frac{\ell}{\beta\,k\log n})$ 轮。我们还提出了一种随机算法，其初始化时间为 $O( \lceil\frac{n}{\beta\,k}\rceil\log n )$ 轮，同时实现了与 $n$ 无关的更新时间：具体而言，针对必须预先固定所有更新的遗忘对手，更新时间为 $O( \lceil \frac{\ell}{\beta\,k} \rceil \log(\beta\,k))$ 轮；若考虑更强的自适应对手，更新时间则为 $O( \lceil \frac{\ell}{\sqrt{\beta\,k}}\rceil \log(\beta\,k))$ 轮。