Abstract
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider this problem in the setting of local algorithms: one wants to quickly determine whether a given edge e is in a specific spanning tree, without computing the whole spanning tree, but rather by inspecting the local neighborhood of e. The challenge is to maintain consistency. That is, to answer queries about different edges according to the same spanning tree. Since it is known that this problem cannot be solved without essentially viewing all the graph, we consider the relaxed version of finding a spanning subgraph with (1+c)n edges instead of n1 edges (where n is the number of vertices and c is a given approximation/sparsity parameter).
It is known that this relaxed problem requires inspecting order of n^{1/2} edges in general graphs (for any constant c), which motivates the study of natural restricted families of graphs. One such family is the family of graphs with an excluded minor (which in particular includes planar graphs). For this family there is an algorithm that achieves constant success probability, and inspects (d/c)^{poly(h)log(1/c)} edges (for each edge it is queried on), where d is the maximum degree in the graph and h is the size of the excluded minor. The distances between pairs of vertices in the spanning subgraph G' are at most a factor of poly(d, 1/c, h) larger than in G.
In this work, we show that for an input graph that is Hminor free for any H of size h, this task can be performed by inspecting only poly(d, 1/c, h) edges in G.
The distances between pairs of vertices in the spanning subgraph G' are at most a factor of h log(d)/c (up to polylogarithmic factors) larger than in G. Furthermore, the error probability of the new algorithm is significantly improved to order of 1/n. This algorithm can also be easily adapted to yield an efficient algorithm for the distributed (message passing) setting.
BibTeX  Entry
@InProceedings{levi_et_al:LIPIcs:2016:6661,
author = {Reut Levi and Dana Ron and Ronitt Rubinfeld},
title = {{A Local Algorithm for Constructing Spanners in MinorFree Graphs}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)},
pages = {38:138:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770187},
ISSN = {18688969},
year = {2016},
volume = {60},
editor = {Klaus Jansen and Claire Mathieu and Jos{\'e} D. P. Rolim and Chris Umans},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6661},
URN = {urn:nbn:de:0030drops66613},
doi = {10.4230/LIPIcs.APPROXRANDOM.2016.38},
annote = {Keywords: spanners, sparse subgraphs, local algorithms, excludedminor}
}
Keywords: 

spanners, sparse subgraphs, local algorithms, excludedminor 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016) 
Issue Date: 

2016 
Date of publication: 

06.09.2016 