Solving Sequential Greedy Problems Distributedly with Sub-Logarithmic Energy Cost

We study the awake complexity of graph problems that belong to the class O-LOCAL, which includes a large subset of problems solvable by sequential greedy algorithms, such as $(\Delta+1)$-coloring, maximal independent set, maximal matching, etc. It is known from previous work that, in $n$-node graphs of maximum degree $\Delta$, any problem in the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\log\Delta+\log^\star n)$. In this paper, we show that any problem belonging to the class O-LOCAL can be solved by a deterministic distributed algorithm with awake complexity $O(\sqrt{\log n}\cdot\log^\star n)$. This leads to a polynomial improvement over the state of the art when $\Delta\gg 2^{\sqrt{\log n}}$, e.g., $\Delta=n^\epsilon$ for some arbitrarily small $\epsilon>0$. The key ingredient for achieving our results is the computation of a network decomposition, that uses a small-enough number of colors, in sub-logarithmic time in the Sleeping model, which can be of independent interest.