Node and Edge Averaged Complexities of Local Graph Problems

Abstract: The node-averaged complexity of a distributed algorithm running on a graph $G=(V,E)$ is the average over the times at which the nodes $V$ of $G$ finish their computation and commit to their outputs. We study the node-averaged complexity for some distributed symmetry breaking problems and provide the following results (among others): - The randomized node-averaged complexity of computing a maximal independent set (MIS) in $n$-node graphs of maximum degree $\Delta$ is at least $\Omega\big(\min\big{\frac{\log\Delta}{\log\log\Delta},\sqrt{\frac{\log n}{\log\log n}}\big}\big)$. This bound is obtained by a novel adaptation of the well-known KMW lower bound [JACM'16]. As a side result, we obtain the same lower bound for the worst-case randomized round complexity for computing an MIS in trees -- this essentially answers open problem 11.15 in the book of Barenboim and Elkin and resolves the complexity of MIS on trees up to an $O(\sqrt{\log\log n})$ factor. We also show that, $(2,2)$-ruling sets, which are a minimal relaxation of MIS, have $O(1)$ randomized node-averaged complexity. - For maximal matching, we show that while the randomized node-averaged complexity is $\Omega\big(\min\big{\frac{\log\Delta}{\log\log\Delta},\sqrt{\frac{\log n}{\log\log n}}\big}\big)$, the randomized edge-averaged complexity is $O(1)$. Further, we show that the deterministic edge-averaged complexity of maximal matching is $O(\log^2\Delta + \log^* n)$ and the deterministic node-averaged complexity of maximal matching is $O(\log^3\Delta + \log^* n)$. - Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be $\Theta(\log n)$, even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity $O(\log^* n)$, while keeping the worst-case complexity in $O(\log n)$.