近最优低能耗确定性分布式SSSP及其对拥塞和APSP的影响
A Near-Optimal Low-Energy Deterministic Distributed SSSP with Ramifications on Congestion and APSP
AI总结：
提出一种低能耗确定性分布式精确SSSP算法，运行 $\tilde{O}(n)$ 轮且节点仅唤醒 $poly(\log n)$ 轮，并基于此通过独立求解与随机调度实现近最优APSP。
摘要：
我们提出了一种低能耗确定性分布式算法，能在近最优时间内计算精确的单源最短路径（SSSP）：算法运行 $\tilde{O}(n)$ 轮，且每个节点仅在 $poly(\log n)$ 轮中保持唤醒。当节点未唤醒时，不执行计算或通信且不消耗能量。我们在实现该结果过程中采用的通用方法可视为对 Dijkstra 经典 SSSP 方法的新颖改造，使其适用于分布式环境。注意，由于 Dijkstra 算法本身需要反复计算整个网络中最小距离的未访问节点，因此在分布式环境中效率不高。我们改造后的方法还具有其他意义，具体如下。作为实现上述最终结果的一步，我们得到了一种简单的确定性精确 SSSP 算法，其时间和消息复杂度分别为近最优的 $\tilde{O}(n)$ 和 $\tilde{O}(m)$，且每条边仅通信 $poly(\log n)$ 条消息。因此，通过简单的随机延迟调度，可以针对 $n$ 个源同时运行 $n$ 个实例，在 $\tilde{O}(n)$ 的近最优时间复杂度内计算所有节点对最短路径（APSP）。该算法采用完全不同的方法达到了 Bernstein 和 Nanongkai [STOC 2019] 近期 APSP 算法的复杂度（且更具模块化，即 SSSP 被独立求解）。由于目前唯一的随机性仅存在于调度中，这也向解决确定性 $\tilde{O}(n)$ 时间 APSP 的开放问题迈进了一步。
英文摘要：
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.
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