Linking and Cutting Spanning Trees

TL;DR

This paper presents an efficient Markov chain-based algorithm for uniformly generating spanning trees in graphs, with proven performance on cycle graphs and promising experimental results on general graphs.

Contribution

It introduces a Markov chain method for spanning tree generation and provides analytical and experimental analysis of its convergence properties.

Findings

Outperforms existing algorithms on cycle graphs

Shows quick convergence in experiments on general graphs

Uses fast data structures for efficiency

Abstract

We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle graphs we prove that this approach significantly outperforms existing algorithms. For general graphs we obtain no analytical bounds, but experimental results show that the chain still converges quickly. This yields an efficient algorithm, also due to the use of proper fast data structures. To bound the mixing time of the chain we describe a coupling, which we analyse for cycle graphs and simulate for other graphs.