Finding connected components in a graph is a fundamental problem in graph analysis. In this work, we present a novel minimum-mapping based Contour algorithm to efficiently solve the connectivity problem. We prove that the Contour algorithm with two or higher order operators can identify all connected components of an undirected graph within $\mathcal{O}(\log d_{max})$ iterations, with each iteration involving $\mathcal{O}(m)$ work, where $d_{max}$ represents the largest diameter among all components in the given graph, and $m$ is the total number of edges in the graph. Importantly, each iteration is highly parallelizable, making use of the efficient minimum-mapping operator applied to all edges. To further enhance its practical performance, we optimize the Contour algorithm through asynchronous updates, early convergence checking, eliminating atomic operations, and choosing more efficient mapping operators. Our implementation of the Contour algorithm has been integrated into the open-source framework Arachne. Arachne extends Arkouda for large-scale interactive graph analytics, providing a Python API powered by the high-productivity parallel language Chapel. Experimental results on both real-world and synthetic graphs demonstrate the superior performance of our proposed Contour algorithm compared to state-of-the-art large-scale parallel algorithm FastSV and the fastest shared memory algorithm ConnectIt. On average, Contour achieves a speedup of 7.3x and 1.4x compared to FastSV and ConnectIt, respectively. All code for the Contour algorithm and the Arachne framework is publicly available on GitHub ( https://github.com/Bears-R-Us/arkouda-njit ), ensuring transparency and reproducibility of our work.

翻译：在图分析中，寻找连通分量是一个基础性问题。本文提出一种基于最小映射的新型轮廓算法（Contour），高效求解图的连通性问题。我们证明：采用二阶或更高阶算子的轮廓算法可在 $\mathcal{O}(\log d_{max})$ 次迭代内识别无向图的所有连通分量，每次迭代的计算量为 $\mathcal{O}(m)$，其中 $d_{max}$ 表示给定图中所有连通分量的最大直径，$m$ 为图的总边数。关键在于，每次迭代具有高度可并行性，通过为所有边应用高效的最小映射算子实现。为提升实际性能，我们通过异步更新、早期收敛检测、消除原子操作及选择更高效的映射算子对轮廓算法进行优化。该算法实现已集成至开源框架 Arachne 中。Arachne 扩展了用于大规模交互式图分析的 Arkouda，提供基于高产能并行语言 Chapel 的 Python API。在真实世界图与合成图上的实验结果表明：与当前最先进的大规模并行算法 FastSV 及最快共享内存算法 ConnectIt 相比，所提轮廓算法性能显著提升。平均而言，Contour 较 FastSV 和 ConnectIt 分别实现 7.3 倍与 1.4 倍的加速比。轮廓算法及 Arachne 框架的全部代码已在 GitHub 上开源（https://github.com/Bears-R-Us/arkouda-njit），确保本工作的透明性与可复现性。