DC-RST: a highly parallel, divide-and-conquer algorithm for creating random spanning trees on a clique

Abstract

We describe DC-RST, a highly parallel, divide and conquer algorithm for generating a random spanning tree of a complete graph on $n$ vertices such that each edge in the graph is chosen with equal probability. DC-RST parallelizes Wilson's sequential random-walk algorithm which has an expected complexity of $\Theta(n)$ on a complete graph. On a system with 48 cores, on some instances, DC-RST was up to 4X faster when first creating random partitions and up to 20X faster without this sub-step. Although the spanning trees generated by DC-RST are random, unlike those generated by Wilson's algorithm, they are not \textit{uniform} random spanning trees.

The proposed application of DC-RST is network analytics, where we seek to determine whether two distance metrics on a graph with $n$ entities are statistically correlated. DC-RST utilizes parallelization to speed up \textsc{DimeCost}, a sequential linear-time algorithm based on uniform random spanning trees. Although DC-RST does not create \textit{uniform} random spanning trees, our preliminary statistical testing comparing it with \textsc{DimeCost}, indicates that results obtained by DC-RST are reliable.