分布式最大匹配：贪婪是最优的
Distributed Maximal Matching: Greedy is Optimal
摘要：
我们研究分布式算法，找到一个最大的匹配在一个匿名的，边缘着色图。如果边缘用$k$颜色适当着色，则存在在$k-1$同步通信轮中找到最大匹配的平凡贪心算法。目前的工作表明，贪心算法在一般情况下是最佳的：任何算法，找到一个最大的匹配匿名，$k$ -边着色图需要$k-1$轮。如果我们专注于最大程度$Δ$的图，它是已知的，最大匹配可以在$O(Δ+ \log^* k)$轮，和以前的工作意味着一个下界的$Ω(\polylog(Δ) + \log^* k)$轮。我们的工作缩小了上界和下界之间的差距：复杂性是$Θ(Δ+ \log^* k)$轮。据我们所知，这是第一个线性的$Δ$下限的分布式复杂性的经典图的问题。
英文摘要：
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: any algorithm that finds a maximal matching in anonymous, $k$-edge-coloured graphs requires $k-1$ 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.
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