Connected Components in Linear Work and Near-Optimal Time

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 \emph{well-connected} components, we can compute connected components on a PRAM in sub-logarithmic parallel time with \emph{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 \textsc{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 $\lambda$.