Sleeping is Superefficient: MIS in Exponentially Better Awake Complexity

Maximal Independent Set (MIS) is one of the central and most well-studied problems in distributed computing. Even after four decades of intensive research, the best-known (randomized) MIS algorithms take $O(\log{n})$ worst-case rounds on general graphs (where $n$ is the number of nodes), while the best-known lower bound is $\Omega\left(\sqrt{\frac{\log{n}}{\log{\log{n}}}}\right)$ rounds. Breaking past the $O(\log{n})$ worst-case bound or showing stronger lower bounds have been longstanding open problems. Our main contribution is that we show that MIS can be computed in (worst-case) awake complexity of $O(\log \log n)$ rounds that is (essentially) exponentially better compared to the (traditional) round complexity lower bound of $\Omega\left(\sqrt{\frac{\log{n}}{\log{\log{n}}}}\right)$. Specifically, we present the following results. (1) We present a randomized distributed (Monte Carlo) algorithm for MIS that with high probability computes an MIS and has $O(\log\log{n})$-rounds awake complexity. This algorithm has (traditional) {\em round complexity} that is $O(poly(n))$. Our bounds hold in the $CONGEST(O(polylog n))$ model where only $O(polylog n)$ (specifically $O(\log^3 n)$) bits are allowed to be sent per edge per round. (2) We also show that we can drastically reduce the round complexity at the cost of a slight increase in awake complexity by presenting a randomized MIS algorithm with $O(\log \log n \log^* n )$ awake complexity and $O(\log^3 n \log \log n \log^*n)$ round complexity in the $CONGEST(O(polylog n))$ model.