Constructing Near Spanning Trees with Few Local Inspections

Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. Motivated by several recent studies of local graph algorithms, we consider the following variant of this problem. Let G be a connected bounded-degree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph G consisting of (1 + ǫ)n edges (for a prespecified constant ǫ > 0), where the decision for different edges should be consistent with the same subgraph G. Can this task be performed by inspecting only a constant number of edges in G? Our main results are: • We show that if every t-vertex subgraph of G has expansion 1/(log t) then one can (deterministically) construct a sparse spanning subgraph G of G using few inspections. To this end we analyze a “local” version of a famous minimum-weight spanning tree algorithm. • We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3-regular graphs of high girth, in which every t-vertex subgraph has expansion 1/(log t).