Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning subgraph containing at most (1+ϵ)n edges (where n is the number of vertices and ϵ is a given approximation/sparsity parameter). In the local setting, the goal is to quickly determine whether a given edge e belongs to such a subgraph, without constructing the whole subgraph, but rather by inspecting (querying) the local neighborhood of e. The challenge ismore »
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 boundeddegree graph. Given an edge e in G we would like to decide whether e belongs to a connected subgraph math formula consisting of math formula edges (for a prespecified constant math formula), where the decision for different edges should be consistent with the same subgraph math formula. Can this task be performed by inspecting only a constant number of edges in G? Our main results are:
We show that if every tvertex subgraph of G has expansion math formula then one can (deterministically) construct a sparse spanning subgraph math formula of G using few inspections. To this end we analyze a “local” version of a famous minimumweight spanning tree algorithm.
We show that the above expansion requirement is sharp even when allowing randomization. To this end we construct a family of 3regular graphs of high girth, in which every tvertex subgraph has expansion math formula. We prove that for this family of graphs, any local algorithm for the sparse spanning graph problem more »
 Award ID(s):
 1650733
 Publication Date:
 NSFPAR ID:
 10026350
 Journal Name:
 Random structures & algorithms
 Volume:
 50
 Issue:
 2
 Page Range or eLocationID:
 183200
 ISSN:
 10982418
 Sponsoring Org:
 National Science Foundation
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