Chatterjee, Gmyr, and Pandurangan [PODC 2020] recently introduced the notion of awake complexity for distributed algorithms, which measures the number of rounds in which a node is awake. In the other rounds, the node is sleeping and performs no computation or communication. Measuring the number of awake rounds can be of significance in many settings of distributed computing, e.g., in sensor networks where energy consumption is of concern. In that paper, Chatterjee et al. provide an elegant randomized algorithm for the Maximal Independent Set (MIS) problem that achieves an $O(1)$ node-averaged awake complexity. That is, the average awake time among the nodes is $O(1)$ rounds. However, to achieve that, the algorithm sacrifices the more standard round complexity measure from the well-known $O(\log n)$ bound of MIS, due to Luby [STOC'85], to $O(\log^{3.41} n)$ rounds. Our first contribution is to present a simple randomized distributed MIS algorithm that, with high probability, has $O(1)$ node-averaged awake complexity and $O(\log n)$ worst-case round complexity. Our second, and more technical contribution, is to show algorithms with the same $O(1)$ node-averaged awake complexity and $O(\log n)$ worst-case round complexity for $(1+\varepsilon)$-approximation of maximum matching and $(2+\varepsilon)$-approximation of minimum vertex cover, where $\varepsilon$ denotes an arbitrary small positive constant.

翻译：Chatterjee、Gmyr 和 Pandurangan [PODC 2020] 近期引入了分布式算法的唤醒复杂度概念，用于衡量节点处于唤醒状态的轮次数量。在其他轮次中，节点处于休眠状态，不执行任何计算或通信。在许多分布式计算场景中（例如需关注能耗的传感器网络），测量唤醒轮次数具有重要价值。在该论文中，Chatterjee 等人提出了一种优雅的随机算法，用于解决最大独立集（MIS）问题，该算法实现了 $O(1)$ 的节点平均唤醒复杂度，即节点间平均唤醒时间为 $O(1)$ 轮。然而，为实现这一目标，该算法将更常规的轮复杂度从 Luby [STOC'85] 著名的 MIS 问题 $O(\log n)$ 界牺牲至 $O(\log^{3.41} n)$ 轮。我们的第一个贡献是提出一种简单的随机化分布式 MIS 算法，该算法以高概率实现 $O(1)$ 节点平均唤醒复杂度和 $O(\log n)$ 最坏情况轮复杂度。我们的第二个且更具技术性的贡献是：针对 $(1+\varepsilon)$ 近似最大匹配和 $(2+\varepsilon)$ 近似最小顶点覆盖问题，展示了具有相同 $O(1)$ 节点平均唤醒复杂度和 $O(\log n)$ 最坏情况轮复杂度的算法，其中 $\varepsilon$ 表示任意小的正常数。