Sleeping is Efficient: MIS in $O(1)$-rounds Node-averaged Awake Complexity

Maximal Independent Set (MIS) is one of the fundamental problems in distributed computing. The round (time) complexity of distributed MIS has traditionally focused on the \emph{worst-case time} for all nodes to finish. The best-known (randomized) MIS algorithms take $O(\log n)$ worst-case rounds on general graphs (where $n$ is the number of nodes). Motivated by the goal to reduce \emph{total} energy consumption in energy-constrained networks such as sensor and ad hoc wireless networks, we take an alternative approach to measuring performance. We focus on minimizing the total (or equivalently, the \emph{average}) time for all nodes to finish. It is not clear whether the currently best-known algorithms yield constant-round (or even $o(\log n)$) node-averaged round complexity for MIS in general graphs. We posit the \emph{sleeping model}, a generalization of the traditional model, that allows nodes to enter either sleep'' or waking’’ states at any round. While waking state corresponds to the default state in the traditional model, in sleeping state a node is ``offline’', i.e., it does not send or receive messages (and messages sent to it are dropped as well) and does not incur any time, communication, or local computation cost. Hence, in this model, only rounds in which a node is awake are counted and we are interested in minimizing the average as well as the worst-case number of rounds a node spends in the awake state. Our main result is that we show that {\em MIS can be solved in (expected) $O(1)$ rounds under node-averaged awake complexity measure} in the sleeping model. In particular, we present a randomized distributed algorithm for MIS that has expected {\em $O(1)$-rounds node-averaged awake complexity} and, with high probability has {\em $O(\log n)$-rounds worst-case awake complexity} and {\em $O({\log }^{3.41}n)$-rounds worst-case complexity}.