Generating random spanning trees more quickly than the cover time

Introduction It is widely known how to generate random spanning trees of an undirected graph. Broder showed how at FOCS 6, and Aldous too found the algorithm 2. Start at any vertex and do a simple random walk on the graph. Each time a vertex is first encountered, mark the edge from which it was discovered. When all the vertices are discovered, the marked edges form a random spanning tree. This algorithm is easy to code up, has small running time constants, and has a nice proof that it generates