Local Algorithms for Sparse Spanning Graphs

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...

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Vydané v:Algorithmica Ročník 82; číslo 4; s. 747 - 786
Hlavní autori: Levi, Reut, Ron, Dana, Rubinfeld, Ronitt
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: New York Springer US 01.04.2020
Springer Nature B.V
Predmet:
Algorithm Analysis and Problem Complexity
Algorithms
Analogies
Apexes
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Graphs
Lower bounds
Mathematics of Computing
Theory of Computation
Local algorithms
Minor-free graphs
Bounded degree model
Sublinear algorithms
Sparse spanning subgraphs
Property testing
ISSN:0178-4617, 1432-0541
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Shrnutí: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 is to maintain consistency. That is, to provide answers concerning different edges according to the same spanning subgraph. We first show that for general bounded-degree graphs, the query complexity of any such algorithm must be Ω ( n ) . This lower bound holds for constant-degree graphs that have high expansion. Next we design an algorithm for (bounded-degree) graphs with high expansion, obtaining a result that roughly matches the lower bound. We then turn to study graphs that exclude a fixed minor (and are hence non-expanding). We design an algorithm for such graphs, which may have an unbounded maximum degree. The query complexity of this algorithm is poly ( 1 / ϵ , h ) (independent of n and the maximum degree), where h is the number of vertices in the excluded minor. Though our two algorithms are designed for very different types of graphs (and have very different complexities), on a high-level there are several similarities, and we highlight both the similarities and the differences.
Bibliografia:ObjectType-Article-1
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content type line 14
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-019-00612-6