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）模型中的连通分量发现问题。当要求总空间与输入图规模呈线性关系时，我们证明：森林图可在常数轮内以高概率求解，一般图则在期望的$2^{O(\log^* n)}$轮内完成，这改进了现有$O(\log \log_{m/n} n)$轮算法。对于目标轮数为常数的情况，我们证明两类问题均可在每轮期望总空间$\Theta(m + n \log^{(k)} n)$内求解，其中$k$为任意大常数，$\log^{(k)}$表示$\log_2$函数的$k$次迭代。相比现有需$\Omega(m + n \log n)$总空间的算法，本方法取得显著改进。