Connected Components in Linear Work and Near-Optimal Time

2023-12-04

arXiv.org

AI Summary

For the connected components (CC) problem on the PRAM model, this paper achieves the first sublogarithmic parallel time with linear total work $O(m+n)$, attaining joint optimality. Methodologically, it introduces the first explicit algorithm whose runtime depends on the graph’s smallest nontrivial eigenvalue (spectral gap) $lambda$: for well-connected graphs with large $lambda$, it computes CCs in $O(log(1/lambda) + loglog n)$ time with high probability—without prior knowledge of $lambda$. This bound is proven tight under the 2-Cycle conjecture. The approach integrates spectral graph theory, randomized contraction, and refined probabilistic analysis, and its optimality is established via a lower-bound reduction. Compared to prior work (e.g., SPAA’20), our algorithm breaks the time bottleneck while preserving linear work, establishing a new paradigm for parallel connectivity detection in spectrally sparse graphs.

Abstract

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 extsc{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 $lambda$.

Problem

• PRAM
• Optimal Workload
• Connected Components

Innovation

• PRAM Model
• Connectivity Algorithm
• Optimal Workload