Maximal Independent Set (MIS) is one of the fundamental and most well-studied problems in distributed graph algorithms. Even after four decades of intensive research, the best-known (randomized) MIS algorithms have $O(\log{n})$ round complexity on general graphs [Luby, STOC 1986] (where $n$ is the number of nodes), while the best-known lower bound is $\Omega(\sqrt{\log{n}/\log\log{n}})$ [Kuhn, Moscibroda, Wattenhofer, JACM 2016]. Breaking past the $O(\log{n})$ round complexity upper bound or showing stronger lower bounds have been longstanding open problems. Our main contribution is to show that MIS can be computed in awake complexity that is exponentially better compared to the best known round complexity of $O(\log n)$ and also bypassing its fundamental $\Omega(\sqrt{\log{n}/\log\log{n}})$ round complexity lower bound exponentially. Specifically, we show that MIS can be computed by a randomized distributed (Monte Carlo) algorithm in $O(\log\log{n} )$ awake complexity with high probability (i.e., with probability at least $1 - n^{-1}$). This algorithm has a round complexity of $O((\log^7 n) \log \log n)$. We also show that we can improve the round complexity at the cost of a slight increase in awake complexity, by presenting a randomized distributed (Monte Carlo) algorithm for MIS that, with high probability, computes an MIS in $O((\log\log{n})\log^*n)$ awake complexity and $O((\log^3 n) (\log \log n) \log^*n)$ round complexity. Our algorithms work in the CONGEST model where messages of size $O(\log n)$ bits can be sent per edge per round.

翻译：最大独立集（MIS）是分布式图算法中最基础且研究最深入的问题之一。即便经过四十年的密集研究，已知最优的（随机化）MIS 算法在一般图上的轮复杂度为 $O(\log{n})$ [Luby, STOC 1986]（其中 $n$ 为节点数），而已知最优下界为 $\Omega(\sqrt{\log{n}/\log\log{n}})$ [Kuhn, Moscibroda, Wattenhofer, JACM 2016]。突破 $O(\log{n})$ 轮复杂度上限或证明更强的下界一直是长期悬而未决的问题。我们的主要贡献在于证明：MIS 的清醒复杂度可指数级优于已知最优的 $O(\log n)$ 轮复杂度，并同样指数级绕过其 $\Omega(\sqrt{\log{n}/\log\log{n}})$ 的轮复杂度基本下界。具体而言，我们证明 MIS 可通过随机化分布式（蒙特卡洛）算法以高概率（即概率至少为 $1 - n^{-1}$）在 $O(\log\log{n} )$ 清醒复杂度内完成计算。该算法的轮复杂度为 $O((\log^7 n) \log \log n)$。此外，我们展示可通过略微增加清醒复杂度来优化轮复杂度：提出一种随机化分布式（蒙特卡洛）MIS 算法，能以高概率在 $O((\log\log{n})\log^*n)$ 清醒复杂度和 $O((\log^3 n) (\log \log n) \log^*n)$ 轮复杂度内计算 MIS。我们的算法工作于 CONGEST 模型，其中每条边每轮可发送 $O(\log n)$ 比特大小的消息。