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+E)n edges (for a prespecified constant E>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/(logt)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/(logt)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. (c) 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 183-200, 2017