In this note, we revisit the recursive random contraction algorithm of Karger and Stein for finding a minimum cut in a graph. Our revisit is occasioned by a paper of Fox, Panigrahi, and Zhang which gives an extension of the Karger-Stein algorithm to minimum cuts and minimum $k$-cuts in hypergraphs. When specialized to the case of graphs, the algorithm is somewhat different than the original Karger-Stein algorithm. We show that the analysis becomes particularly clean in this case: we can prove that the probability that a fixed minimum cut in an $n$ node graph is returned by the algorithm is bounded below by $1/(2H_n-2)$, where $H_n$ is the $n$th harmonic number. We also consider other similar variants of the algorithm, and show that no such algorithm can achieve an asymptotically better probability of finding a fixed minimum cut.
翻译:在本说明中,我们重新审视了Karger和Stein的递归随机收缩算法,以在图表中找到最小的切分。我们的重访是由Fox、Panigrahi和Zhang的论文引发的,该文件将Karger-Stein算法的扩展范围扩大到最小的削减和高音中最小的折算法。在对图表进行专门研究时,算法与最初的Karger-Stein算法略有不同。我们显示,在这种情况下,分析变得特别干净:我们可以证明,在美元节点图中固定的最小切分率由1美元/(2H_n-2)返回的概率是低于1美元/(2H_n)美元,而美元是第1美元调和数。我们还考虑了其他类似的算法变量,并表明,任何这种算法都不可能在寻找固定的最低切分法方面获得同样更好的概率。