The shortest path problem is a typical problem in graph theory with wide potential applications. The state-of-the-art single-source shortest paths algorithm on the weight graph is the $\Delta$-stepping algorithm, which can efficiently process weighted graphs in parallel. DAWN is an algorithm that addresses the shortest path problem on unweighted graphs, and we propose a weighted version that can handle graphs with weights edges, while maintaining the high scalability and parallelism features as DAWN. The novel version requires $O(\mu m)$ and $O(\mu \cdot E_{wcc})$ times on the connected and unconnected graphs for SSSP problems, respectively. $E_{wcc}$ denote the number of edges included in the largest weakly connected component, and $\mu$ is a constant denoting the average number of path transformations in the tasks. We tested the weighted version on the real graphs from Stanford Network Analysis Platform and SuiteSparse Matrix Collection, which outperformed the solution of $\Delta$-stepping algorithm from Gunrock, achieving a speedup of 43.163$\times$.

翻译：最短路径问题是图论中的经典问题，具有广泛的潜在应用。在加权图上，当前最先进的单源最短路径算法是$\Delta$-stepping算法，该算法能够高效地并行处理加权图。DAWN是一种解决无权图最短路径问题的算法，我们提出其加权版本，可处理带权边图，同时保留DAWN的高可扩展性和并行性。该新版本对于SSSP问题在连通图和未连通图上的时间复杂度分别为$O(\mu m)$和$O(\mu \cdot E_{wcc})$。其中$E_{wcc}$表示最大弱连通分量中包含的边数，$\mu$为常数，表示任务中路径变换的平均次数。我们在斯坦福网络分析平台和SuiteSparse矩阵集合的真实图上测试了该加权版本，其性能优于基于Gunrock的$\Delta$-stepping算法解决方案，加速比达43.163倍。