We study the problem of finding connected components in the Adaptive Massively Parallel Computation (AMPC) model. We show that when we require the total space to be linear in the size of the input graph the problem can be solved in $O(\log^* n)$ rounds in forests (with high probability) and $2^{O(\log^* n)}$ expected rounds in general graphs. This improves upon an existing $O(\log \log_{m/n} n)$ round algorithm. For the case when the desired number of rounds is constant we show that both problems can be solved using $\Theta(m + n \log^{(k)} n)$ total space in expectation (in each round), where $k$ is an arbitrarily large constant and $\log^{(k)}$ is the $k$-th iterate of the $\log_2$ function. This improves upon existing algorithms requiring $\Omega(m + n \log n)$ total space.

翻译：我们研究自适应大规模并行计算（AMPC）模型中寻找连通分量的问题。我们证明，当要求总空间与输入图规模呈线性关系时，该问题可在森林上以高概率在$O(\log^* n)$轮内解决，在一般图上以$2^{O(\log^* n)}$期望轮数内解决。这改进了现有的$O(\log \log_{m/n} n)$轮算法。对于所需轮数为常数的情况，我们证明这两种问题均可使用$\Theta(m + n \log^{(k)} n)$的总期望空间（每轮）解决，其中$k$为任意大的常数，$\log^{(k)}$表示$\log_2$函数的$k$次迭代。这改进了现有需要$\Omega(m + n \log n)$总空间的算法。