We present a low-energy deterministic distributed algorithm that computes exact Single-Source Shortest Paths (SSSP) in near-optimal time: it runs in $\tilde{O}(n)$ rounds and each node is awake during only $poly(\log n)$ rounds. When a node is not awake, it performs no computations or communications and spends no energy. The general approach we take along the way to this result can be viewed as a novel adaptation of Dijkstra's classic approach to SSSP, which makes it suitable for the distributed setting. Notice that Dijkstra's algorithm itself is not efficient in the distributed setting due to its need for repeatedly computing the minimum-distance unvisited node in the entire network. Our adapted approach has other implications, as we outline next. As a step toward the above end-result, we obtain a simple deterministic algorithm for exact SSSP with near-optimal time and message complexities of $\tilde{O}(n)$ and $\tilde{O}(m)$, in which each edge communicates only $poly(\log n)$ messages. Therefore, one can simultaneously run $n$ instances of it for $n$ sources, using a simple random delay scheduling. That computes All Pairs Shortest Paths (APSP) in the near-optimal time complexity of $\tilde{O}(n)$. This algorithm matches the complexity of the recent APSP algorithm of Bernstein and Nanongkai [STOC 2019] using a completely different method (and one that is more modular, in the sense that the SSSPs are solved independently). It also takes a step toward resolving the open problem on a deterministic $\tilde{O}(n)$-time APSP, as the only randomness used now is in the scheduling.

翻译：我们提出了一种低能耗确定性分布式算法，该算法能以近最优时间计算精确的单源最短路径：算法运行时间为$\tilde{O}(n)$轮，且每个节点仅在$poly(\log n)$轮内保持活跃状态。当节点处于非活跃状态时，其不执行任何计算或通信操作，也不消耗能量。为实现这一结果所采用的整体方法可视为对Dijkstra经典SSSP算法的一种新颖适配，使其适用于分布式环境。值得注意的是，由于Dijkstra算法需要反复计算整个网络中距离最小的未访问节点，其本身在分布式环境中并不高效。我们提出的适配方法还具有其他影响，具体如下。作为实现上述最终结果的步骤，我们首先获得了一个简单的确定性精确SSSP算法，其具有近最优的时间复杂度$\tilde{O}(n)$和消息复杂度$\tilde{O}(m)$，且每条边仅传输$poly(\log n)$条消息。因此，通过简单的随机延迟调度机制，可以同时运行该算法的$n$个实例以处理$n$个源节点。这能以$\tilde{O}(n)$的近最优时间复杂度计算全对最短路径。该算法与Bernstein和Nanongkai近期提出的APSP算法[STOC 2019]具有相同的复杂度，但采用了完全不同的方法（且更具模块化特性，因为各SSSP实例是独立求解的）。该研究也为解决确定性$\tilde{O}(n)$时间APSP算法的开放问题迈出了重要一步，因为当前算法中仅调度环节使用了随机性。