Computing the connected components of a graph is a fundamental problem in algorithmic graph theory. A major question in this area is whether we can compute connected components in $o(\log n)$ parallel time. Recent works showed an affirmative answer in the Massively Parallel Computation (MPC) model for a wide class of graphs. Specifically, Behnezhad et al. (FOCS'19) showed that connected components can be computed in $O(\log d + \log \log n)$ rounds in the MPC model. More recently, Liu et al. (SPAA'20) showed that the same result can be achieved in the standard PRAM model but their result incurs $\Theta((m+n) \cdot (\log d + \log \log n))$ work which is sub-optimal. In this paper, we show that for graphs that contain well-connected components, we can compute connected components on a PRAM in sub-logarithmic parallel time with optimal, i.e., $O(m+n)$ total work. Specifically, our algorithm achieves $O(\log(1/\lambda) + \log \log n)$ parallel time with high probability, where $\lambda$ is the minimum spectral gap of any connected component in the input graph. The algorithm requires no prior knowledge on $\lambda$. Additionally, based on the 2-Cycle Conjecture we provide a time lower bound of $\Omega(\log(1/\lambda))$ for solving connected components on a PRAM with $O(m+n)$ total memory when $\lambda \le (1/\log n)^c$, giving conditional optimality to the running time of our algorithm as a parameter of $\lambda$.

翻译：计算图的连通分量是算法图论中的一个基本问题。该领域的一个核心问题是：能否在$o(\log n)$并行时间内完成连通分量计算？近期研究表明，在针对广泛图类的大规模并行计算（MPC）模型中，该问题得到了肯定答案。具体而言，Behnezhad等人（FOCS'19）证明了在MPC模型下，连通分量可在$O(\log d + \log \log n)$轮内完成计算。随后，Liu等人（SPAA'20）表明该结果可在标准PRAM模型中复现，但其方法需$\Theta((m+n) \cdot (\log d + \log \log n))$工作量，未达到最优。本文证明：对于具备良好连通性的图结构，我们可在亚对数级并行时间内，以最优总工作量$O(m+n)$完成PRAM上的连通分量计算。具体而言，我们的算法以高概率实现$O(\log(1/\lambda) + \log \log n)$并行时间，其中$\lambda$为输入图中任意连通分量的最小谱隙。该算法无需预先知晓$\lambda$值。此外，基于2-循环猜想，我们给出了当$\lambda \le (1/\log n)^c$时，在总内存为$O(m+n)$的PRAM上求解连通分量问题的时间下界$\Omega(\log(1/\lambda))$，从而证明了算法运行时间关于参数$\lambda$的条件最优性。