Node and Edge Averaged Complexities of Local Graph Problems

Alkida Balliu, Mohsen Ghaffari, Fabian Kuhn, Dennis Olivetti
2022 Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing  
We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. 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 of the central distributed symmetry breaking problems. As our main result, we show that the randomized node-averaged complexity of
more » ... mputing a maximal independent set (MIS) is at least Ω(min{log Δ/log log Δ, √︁ log 𝑛/log log 𝑛}) in 𝑛-node graphs of maximum degree Δ. This bound is obtained by a novel adaptation of the well-known lower bound of Kuhn, Moscibroda, and Wattenhofer [JACM'16]. As a side result, we obtain that the worstcase randomized round complexity for computing an MIS in trees is also Ω(min{log Δ/log log Δ, √︁ log 𝑛/log log 𝑛})-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 ( √︁ log log 𝑛) factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to 𝑂 (1). For the problem of computing a maximal matching, we show that while the randomized node-averaged complexity is at least Ω(min{log Δ/log log Δ, √︁ log 𝑛/log log 𝑛}), the randomized edgeaveraged complexity is 𝑂 (1). Further, we show that the deterministic edge-averaged complexity of maximal matching is 𝑂 (log 2 Δ + log * 𝑛) and the deterministic node-averaged complexity of maximal matching is 𝑂 (log 3 Δ + 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 Θ(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).
doi:10.1145/3519270.3538419 fatcat:xz4l2ozpgrfxzbxmspaadszd6e