A Local Algorithm for Constructing Spanners in Minor-Free Graphs

Authors Reut Levi, Dana Ron, Ronitt Rubinfeld

Thumbnail PDF


  • Filesize: 0.57 MB
  • 15 pages

Document Identifiers

Author Details

Reut Levi
Dana Ron
Ronitt Rubinfeld

Cite AsGet BibTex

Reut Levi, Dana Ron, and Ronitt Rubinfeld. A Local Algorithm for Constructing Spanners in Minor-Free Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 38:1-38:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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.
  • spanners
  • sparse subgraphs
  • local algorithms
  • excluded-minor


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7(4):567-583, 1986. URL: http://dx.doi.org/10.1016/0196-6774(86)90019-2.
  2. N. Alon, R. Rubinfeld, S. Vardi, and N. Xie. Space-efficient local computation algorithms. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1132-1139, 2012. Google Scholar
  3. Noga Alon, Paul D. Seymour, and Robin Thomas. A separator theorem for graphs with an excluded minor and its applications. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 13-17, 1990, Baltimore, Maryland, USA, pages 293-299, 1990. URL: http://dx.doi.org/10.1145/100216.100254.
  4. G. Even, M. Medina, and D. Ron. Deterministic stateless centralized local algorithms for bounded degree graphs. In Algorithms - ESA 2014 - 22th Annual European Symposium, Wroclaw, Poland, September 8-10, 2014. Proceedings, pages 394-405, 2014. URL: http://dx.doi.org/10.1007/978-3-662-44777-2_33.
  5. Ken-ichi Kawarabayashi, Philip N. Klein, and Christian Sommer. Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part I, pages 135-146, 2011. URL: http://dx.doi.org/10.1007/978-3-642-22006-7_12.
  6. Ken-ichi Kawarabayashi and Bruce A. Reed. A separator theorem in minor-closed classes. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 153-162, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.22.
  7. R. Levi and D. Ron. A quasi-polynomial time partition oracle for graphs with an excluded minor. ACM Trans. Algorithms, 11(3):24:1-24:13, 2015. Google Scholar
  8. R. Levi, D. Ron, and R. Rubinfeld. Local algorithms for sparse spanning graphs. In Proceedings of the Eighteenth International Workshop on Randomization and Computation (RANDOM), pages 826-842, 2014. Google Scholar
  9. Reut Levi, Guy Moshkovitz, Dana Ron, Ronitt Rubinfeld, and Asaf Shapira. Constructing near spanning trees with few local inspections. Random Structures &Algorithms, pages n/a-n/a, 2016. URL: http://dx.doi.org/10.1002/rsa.20652.
  10. W. Mader. Homomorphiesätze für graphen. Mathematische Annalen, 178:154-168, 1968. Google Scholar
  11. Y. Mansour, A. Rubinstein, S. Vardi, and N. Xie. Converting online algorithms to local computation algorithms. In Automata, Languages and Programming: Thirty-Ninth International Colloquium (ICALP), pages 653-664, 2012. Google Scholar
  12. Y. Mansour and S. Vardi. A local computation approximation scheme to maximum matching. In Proceedings of the Sixteenth International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pages 260-273, 2013. Google Scholar
  13. D. Peleg and A. A. Schäffer. Graph spanners. Journal of Graph Theory, 13:99-116, 1989. Google Scholar
  14. D. Peleg and J. D. Ullman. An optimal synchronizer for the hypercube. SIAM Journal on Computing, 18:229-243, 1989. Google Scholar
  15. L. S. Ram and E. Vicari. Distributed small connected spanning subgraph: Breaking the diameter bound. Technical report, Zürich, 2011. Google Scholar
  16. R. Rubinfeld, G. Tamir, S. Vardi, and N. Xie. Fast local computation algorithms. In Proceedings of The Second Symposium on Innovations in Computer Science (ICS), pages 223-238, 2011. Google Scholar