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. Energy is a premium resource in various settings such as battery-powered wireless networks and sensor networks. The bulk of the energy is used by nodes when they are awake, i.e., when they are sending, receiving, and even just listening for messages. On the other hand, when a node is sleeping, it does not perform any communication and thus spends very little energy. Several recent works have addressed the problem of designing energy-efficient distributed algorithms for various fundamental problems. These algorithms operate by minimizing the number of rounds in which any node is awake, also called the (worst-case) awake complexity. An intriguing open question is whether one can design a distributed MIS algorithm that has significantly smaller awake complexity compared to existing algorithms. In particular, the question of obtaining a distributed MIS algorithm with \(o(\log n)\) awake complexity was left open in [Chatterjee, Gmyr, Pandurangan, PODC 2020]. 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 \(\mathcal{CONGEST}\) model where messages of size \(O(\log n)\) bits can be sent per edge per round.
