Generating Random Spanning Trees via Fast Matrix Multiplication.

We consider the problem of sampling a uniformly random spanning tree of a graph. This is a classic algorithmic problem for which several exact and approximate algorithms are known. Random spanning trees have several connections to Laplacian matrices; this leads to algorithms based on fast matrix multiplication. The best algorithm for dense graphs can produce a uniformly random spanning tree of an n-vertex graph in time (O(n^{2.38})). This algorithm is intricate and requires explicitly computing the LU-decomposition of the Laplacian.