Constructing near spanning trees with few local inspections

Abstract

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+ )n edges (for a prespecified constant > 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.

Non-local Probes Do Not Help with Many Graph Problems

“…On the other hand lower bounds in this model are almost nonexistent. In fact, the only lower bound shown directly in the centralised local model is for the spanning graph problem in which the CentLOCAL algorithm computes a "tree-like" subgraph of a given bounded degree graph [16,17].…”

Decomposing a graph into expanding subgraphs

“…Formally, an ‐local sparse spanning graph algorithm makes at most q queries to the incidence‐lists representation of the (bounded‐degree, connected) input graph G , and provides query access to a connected subgraph of G that has fewer than edges. The question of which graphs have a local spanning graph algorithm that uses only a constant number of queries was studied in , where it was shown that the answer is given by the same hereditary notion of expansion considered in the current paper. A graph is said to be f‐non‐expanding if every t ‐vertex subgraph H has edge expansion at most f ( t ).…”

“…Motivated by the growing literature on local algorithms (see Rubinfeld et al and the references in ), let us consider the problem of constructing a spanning tree of a graph G in a local manner. By this we mean being able to decide if a given edge of G belongs to the tree in constant time (and in particular, without constructing the entire tree).…”

“…The question of which graphs have a local spanning graph algorithm that uses only a constant number of queries was studied in [22], where it was shown that the answer is given by the same hereditary notion of expansion considered in the current paper. A graph is said to be f -non-expanding if every t-vertex subgraph H has edge expansion at most f (t).…”

“…A graph is said to be f -non-expanding if every t-vertex subgraph H has edge expansion at most f (t). It was shown in [22] that if the input graphs are f -non-expanding with f (x) =…”