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 + ϵ ) n edges (for a prespecified constant ϵ > 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 ) 1 + o ( 1 )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 ) 1 − o ( 1 ). We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem requires inspecting a number of edges which is proportional to the girth. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 183–200, 2017