The study of approximate matching in the Massively Parallel Computations (MPC) model has recently seen a burst of breakthroughs. Despite this progress, however, we still have a far more limited understanding of maximal matching which is one of the central problems of parallel and distributed computing. All known MPC algorithms for maximal matching either take polylogarithmic time which is considered inefficient, or require a strictly super-linear space of $n^{1+\Omega(1)}$ per machine. In this work, we close this gap by providing a novel analysis of an extremely simple algorithm a variant of which was conjectured to work by Czumaj et al. [STOC'18]. The algorithm edge-samples the graph, randomly partitions the vertices, and finds a random greedy maximal matching within each partition. We show that this algorithm drastically reduces the vertex degrees. This, among some other results, leads to an $O(\log \log \Delta)$ round algorithm for maximal matching with $O(n)$ space (or even mildly sublinear in $n$ using standard techniques). As an immediate corollary, we get a $2$ approximate minimum vertex cover in essentially the same rounds and space. This is the best possible approximation factor under standard assumptions, culminating a long line of research. It also leads to an improved $O(\log\log \Delta)$ round algorithm for $1 + \varepsilon$ approximate matching. All these results can also be implemented in the congested clique model within the same number of rounds.
翻译:在大规模平行计算模型(MPC)的近似匹配研究中,最近出现了突破。尽管取得了这一进展,我们仍然对最大匹配的理解有限得多,这是平行计算和分布计算的中心问题之一。所有已知最大匹配的 MPC 算法要么需要效率低下的多对数时间,要么需要严格的超级线性空间,每台机器1美元,或者需要1美元1美元Omega(1)美元。在这项工作中,我们通过对极简单的算法进行新的分析来弥补这一差距,一种极简单的算法变量被Czumaj 和 Al. [STOC'18] 所推测为最大匹配。算法边缘对图进行抽样抽样抽样,随机分割顶端,发现每个分区内随机贪婪的最大匹配。我们发现,这种算法会大幅降低顶点的温度。除了其他一些结果外,这可以导致美元(log\log\Delta) 的所有圆算法与美元(美元)的基数一致,一个比对美元空间的变法值(甚至轻微的亚值 美元,在美元基底的底底底线上,在标准的基底基数中也可以得到一个最接近的基的基的基的计算。