In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown {\em weighted} undirected graph making $O(n)$ $\mathsf{CUT}$ queries in expectation. For weighted graphs, this is optimal due to a result in [Auza and Lee, 2021] which shows an $\Omega(n)$ lower bound for zero-error randomized algorithms. %To our knowledge, it is the only regime of this problem where we have upper and lower bounds tight up to constants. These questions have been extensively studied in the past few years, especially due to the problem's connections to symmetric submodular function minimization. We also describe a simple polynomial time deterministic algorithm that makes $O(\frac{n\log n}{\log\log n})$ queries on undirected unweighted graphs and returns a maximal spanning forest, thereby (slightly) improving upon the state-of-the-art.

翻译：本文描述了一种随机算法，该算法通过期望次数为$O(n)$的$\mathsf{CUT}$查询，返回未知加权无向图的最大生成森林。对于加权图，由于[Auza and Lee, 2021]中证明的零误差随机算法存在$\Omega(n)$下界，该结果是渐进最优的。我们还描述了一个简单的多项式时间确定性算法，该算法在无向无权重图上进行$O(\frac{n\log n}{\log\log n})$次查询，并返回最大生成森林，从而（略微）改进了现有最优结果。