We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle graphs we prove that this approach significantly outperforms existing algorithms. For general graphs we obtain no analytical bounds, but experimental results show that the chain still converges quickly. This yields an efficient algorithm, also due to the use of proper fast data structures. To bound the mixing time of the chain we describe a coupling, which we analyse for cycle graphs and simulate for other graphs.

中文翻译：

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链接和切割生成树

我们考虑统一生成连通无向图的生成树的问题。这个过程对于计算统计数据很有用，即对于系统发育树。我们描述了用于生成这些树的马尔可夫链。对于循环图，我们证明这种方法明显优于现有算法。对于一般的图，我们没有得到解析界限，但实验结果表明链仍然快速收敛。这产生了一种有效的算法，这也是由于使用了适当的快速数据结构。为了限制链的混合时间，我们描述了一个耦合，我们分析循环图并模拟其他图。