We study distributed algorithms that find a maximal matching in an anonymous, edge-coloured graph. If the edges are properly coloured with k colours, there is a trivial greedy algorithm that finds a maximal matching in k-1 synchronous communication rounds. The present work shows that the greedy algorithm is optimal in the general case: if A is a deterministic distributed algorithm that finds a maximal matching in anonymous, k-edge-coloured graphs, then there is a worst-case input in which the running time of A is at least k1 rounds. If we focus on graphs of maximum degree Δ, it is known that a maximal matching can be found in O(Δ+ log* k) rounds, and prior work implies a lower bound of Ω(polylog(Δ) + log* k) rounds. Our work closes the gap between upper and lower bounds: the complexity is Θ(Δ+ log* k) rounds. To our knowledge, this is the first linear-in-Δ lower bound for the distributed complexity of a classical graph problem.
It is easy to come up with an artificial problem with the complexity of @math , but so far no such tight results are known for classical graph problems such as maximal matchings. The lower-bound result by Kuhn and Wattenhofer @cite_3 comes close, but it only applies to a restricted family of algorithms.
@cite_3
2011595776
We show that maximal matchings can be computed deterministically in O(log4 n) rounds in the synchronous, message-passing model of computation. This is one of the very few cases known of a nontrivial graph structure, and the only "classical" one, which can be computed distributively in polylogarithmic time without recourse to randomization.