A Local Algorithm for Constructing Spanners in Minor-Free Graphs

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 n-1 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 H-minor 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 poly-logarithmic 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.

[1]  David Peleg,et al.  An Optimal Synchronizer for the Hypercube , 1989, SIAM J. Comput..

[2]  Ronitt Rubinfeld,et al.  Constructing near spanning trees with few local inspections , 2015, Random Struct. Algorithms.

[3]  Yishay Mansour,et al.  Converting Online Algorithms to Local Computation Algorithms , 2012, ICALP.

[4]  Jose Augusto Ramos Soares,et al.  Graph Spanners: a Survey , 1992 .

[5]  Ronitt Rubinfeld,et al.  Local Algorithms for Sparse Spanning Graphs , 2014, APPROX-RANDOM.

[6]  Yishay Mansour,et al.  A Local Computation Approximation Scheme to Maximum Matching , 2013, APPROX-RANDOM.

[7]  W. Mader Homomorphiesätze für Graphen , 1968 .

[8]  David Peleg,et al.  An optimal synchronizer for the hypercube , 1987, PODC '87.

[9]  Dana Ron,et al.  A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor , 2013, ICALP.

[10]  RonDana,et al.  A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor , 2015 .

[11]  Robin Thomas,et al.  A separator theorem for graphs with an excluded minor and its applications , 1990, STOC '90.

[12]  Ronitt Rubinfeld,et al.  Fast Local Computation Algorithms , 2011, ICS.

[13]  Bruce A. Reed,et al.  A Separator Theorem in Minor-Closed Classes , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[14]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[15]  Ken-ichi Kawarabayashi,et al.  Linear-Space Approximate Distance Oracles for Planar, Bounded-Genus and Minor-Free Graphs , 2011, ICALP.

[16]  Dana Ron,et al.  Deterministic Stateless Centralized Local Algorithms for Bounded Degree Graphs , 2014, ESA.

[17]  Noga Alon,et al.  Space-efficient local computation algorithms , 2011, SODA.