Spanning trees in random satisfiability problems

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

Random generalized satisfiability problems

We develop a coherent model in order to study specific random constraint satisfaction problems. Our framework includes all the generalized satisfiability problems defined by Schaefer. For such problems we study the phase transition. Our model captures problems having a sharp transition as well as problems having a coarse transition. We first give lower and upper bounds for the location of the p...

Random Walks and Random Spanning Trees

We explore algorithms for generating random spanning trees. We first study an algorithm that was developed independently by David Alduous [1] and Andrei Broder [2]. The algorithm uses a a simple random walk in which edges that correspond to the first visit to vertices are added to the spanning tree. Analysis was inspired by Andrei Broder’s paper. Additionally, we study an algorithm by David Wil...

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...