Expanders via Random Spanning Trees

Expanders via random spanning trees

Motivated by the problem of routing reliably and scalably in a graph, we introduce the notion of a splicer, the union of spanning trees of a graph. We prove that for any bounded-degree nvertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n). For the random graph Gn,p, for p = Ω(log n/n), we give a randomized algor...

Generating Random Spanning Trees

This paper describes a probabilistic algorithm that, given a connected, undirected graph G with n vertices, produces a spanning tree of G chosen uniformly at random among the spanning trees of G. The expected running time is O(n logn) per generated tree for almost all graphs, and O(n3) for the worst graphs. Previously known deterministic algorithms and much more complicated and require O(n3) ti...

Generating Random Spanning Trees

Expanders are universal for the class of all spanning trees

Given a class of graphs F , we say that a graph G is universal for F , or F-universal, if every H ∈ F is contained in G as a subgraph. The construction of sparse universal graphs for various families F has received a considerable amount of attention. One is particularly interested in tight F-universal graphs, i. e., graphs whose number of vertices is equal to the largest number of vertices in a...

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 ...