Connected Components in Linear Work and Near-Optimal Time
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912209556209664 |
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| author | Farhadi, Alireza Liu, S. Cliff Shi, Elaine |
| author_facet | Farhadi, Alireza Liu, S. Cliff Shi, Elaine |
| contents | 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 $Θ((m+n) \cdot (\log d + \log \log n))$ 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 $O(\log(1/λ) + \log \log n)$ parallel time with high probability, where $λ$ is the minimum spectral gap of any connected component in the input graph. The algorithm requires no prior knowledge on $λ$.
Additionally, based on the \textsc{2-Cycle} Conjecture we provide a time lower bound of $Ω(\log(1/λ))$ for solving connected components on a PRAM with $O(m+n)$ total memory when $λ\le (1/\log n)^c$, giving conditional optimality to the running time of our algorithm as a parameter of $λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_02332 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Connected Components in Linear Work and Near-Optimal Time Farhadi, Alireza Liu, S. Cliff Shi, Elaine Data Structures and Algorithms F.2.2 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 $Θ((m+n) \cdot (\log d + \log \log n))$ 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 $O(\log(1/λ) + \log \log n)$ parallel time with high probability, where $λ$ is the minimum spectral gap of any connected component in the input graph. The algorithm requires no prior knowledge on $λ$. Additionally, based on the \textsc{2-Cycle} Conjecture we provide a time lower bound of $Ω(\log(1/λ))$ for solving connected components on a PRAM with $O(m+n)$ total memory when $λ\le (1/\log n)^c$, giving conditional optimality to the running time of our algorithm as a parameter of $λ$. |
| title | Connected Components in Linear Work and Near-Optimal Time |
| topic | Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2312.02332 |