There are distributed graph algorithms for finding maximal matchings and maximal independent sets in $O(\Delta + \log^* n)$ communication rounds; here $n$ is the number of nodes and $\Delta$ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on $n$ is optimal: these problems cannot be solved in $o(\log^* n)$ rounds even if $\Delta = 2$. However, the dependency on $\Delta$ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least $1-1/n$ requires $\Omega(\min\{\Delta,\log \log n / \log \log \log n\})$ rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires $\Omega(\min\{\Delta, \log n / \log \log n\})$ rounds; this is an improvement over prior lower bounds also as a function of $n$.
翻译:找到最大匹配和最大独立套件的分布式图表算法以$O (\ Delta +\ log\\ n) n美元为最大匹配和最大独立套件。 这里的美元是节点的数量和$\ Delta$是最大程度。 Linial( 1987, 1992) 所约束的较低值表明对美元的依赖度是最佳的: 这些问题即使$\ Delta = 2美元也无法用$( $\ Delta = 2美元) 解决。 但是, 对$\ Delta 美元的依赖是一个长期的未决问题, 目前上下界之间有指数差距。 我们证明上界是紧的。 我们显示, 任何找到最大匹配或最大独立套件的算法, 概率至少为 1-1/ n美元 ; log nlog\ log n/ log n log n = $ n 。 在分布式的 LOCAL 计算模型中, 对美元的依赖度是一个长期的未决问题, 并且当前在上下界和下界间存在一个指数差距差距。 我们证明, 任何确定性算中找到最高匹配或最大匹配/\ 最大值的值为美元。