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.

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