We present new deterministic algorithms for computing distributed weighted minimum weight cycle (MWC) in undirected and directed graphs and distributed weighted all nodes shortest cycle (ANSC) in directed graphs. Our algorithms for these problems run in $\tilde{O}(n)$ rounds in the CONGEST model on graphs with arbitrary non-negative edge weights, matching the lower bound up to polylogarithmic factors. Before our work, no near linear rounds deterministic algorithms were known for these problems. The previous best bound for solving these problems deterministically requires an initial computation of all pairs shortest paths (APSP) on the given graph, followed by post-processing of $O(n)$ rounds, and in total takes $\tilde{O}(n^{4/3})$ rounds, using deterministic APSP~\cite{AR-SPAA20}. The main component of our new $\tilde{O}(n)$ rounds algorithms is a deterministic technique for constructing a sequence of successive blocker sets. These blocker sets are then treated as source nodes to compute $h$-hop shortest paths, which can then be used to compute candidate shortest cycles whose hop length lies in a particular range. The shortest cycles can then be obtained by selecting the cycle with the minimum weight from all these candidate cycles. Additionally using the above blocker set sequence technique, we also obtain $\tilde{O}(n)$ rounds deterministic algorithm for the multi-source shortest paths problem (MSSP) for both directed and undirected graphs, given that the size of the source set is at most $\sqrt{n}$. This new result for MSSP can be a step towards obtaining a $o(n^{4/3})$ rounds algorithm for deterministic APSP. We also believe that our new blocker set sequence technique may have potential applications for other distributed algorithms.

翻译：本文提出了新的确定性算法，用于在无向图和有向图中计算分布式加权最小权值环（MWC），以及有向图中分布式加权所有结点最短环（ANSC）。针对具有任意非负边权的图，这些算法在CONGEST模型中以$\tilde{O}(n)$轮运行，与下界仅相差多对数因子。在此之前，这些问题尚未有已知的近似线性轮数确定性算法。现有最优确定性解法需先计算给定图的全对最短路径（APSP），再经过$O(n)$轮后处理，总轮数达到$\tilde{O}(n^{4/3})$（基于确定性APSP~\cite{AR-SPAA20}）。我们新$\tilde{O}(n)$轮算法的核心组件是一种用于构建连续阻塞集序列的确定性技术。这些阻塞集被用作源节点计算$h$跳最短路径，进而可计算跳长位于特定范围内的候选最短环。通过从所有候选环中选取最小权值环即可得到最短环。此外，利用上述阻塞集序列技术，我们还得到了源节点集规模不超过$\sqrt{n}$时，有向图和无向图多源最短路径问题（MSSP）的$\tilde{O}(n)$轮确定性算法。这一MSSP新结果可视为实现确定性APSP的$o(n^{4/3})$轮算法的关键步骤。我们相信，该新型阻塞集序列技术在其他分布式算法中也具有潜在应用价值。