A Centralized Local Algorithm for the Sparse Spanning Graph Problem

Authors Christoph Lenzen, Reut Levi

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Author Details

Christoph Lenzen
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Reut Levi
  • Weizmann Institute of Science, Rehovot, Israel

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Christoph Lenzen and Reut Levi. A Centralized Local Algorithm for the Sparse Spanning Graph Problem. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 87:1-87:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Constructing a sparse spanning subgraph is a fundamental primitive in graph theory. In this paper, we study this problem in the Centralized Local model, where the goal is to decide whether an edge is part of the spanning subgraph by examining only a small part of the input; yet, answers must be globally consistent and independent of prior queries. Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees, cannot be constructed efficiently in this model. Therefore, we settle for a spanning subgraph containing at most (1+epsilon)n edges (where n is the number of vertices and epsilon is a given approximation/sparsity parameter). We achieve a query complexity of O~(poly(Delta/epsilon)n^{2/3}), where Delta is the maximum degree of the input graph. Our algorithm is the first to do so on arbitrary bounded degree graphs. Moreover, we achieve the additional property that our algorithm outputs a spanning subgraph of bounded stretch i.e., distances are approximately preserved. With high probability, for each deleted edge there is a path of O(log n * (Delta+log n)/epsilon) hops in the output that connects its endpoints.

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • local
  • spanning graph
  • sparse


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  1. 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
  2. Artur Czumaj, Oded Goldreich, Dana Ron, C. Seshadhri, Asaf Shapira, and Christian Sohler. Finding cycles and trees in sublinear time. Random Structures and Algorithms, 45(2):139-184, 2014. Google Scholar
  3. Michael Elkin and Ofer Neiman. Efficient algorithms for constructing very sparse spanners and emulators. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 652-669, 2017. Google Scholar
  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. U. Feige, Y Mansour, and R. E. Schapire. Learning and inference in the presence of corrupted inputs. In Proceedings of The 28th Conference on Learning Theory, COLT 2015, Paris, France, July 3-6, 2015, pages 637-657, 2015. Google Scholar
  6. Hendrik Fichtenberger, Reut Levi, Yadu Vasudev, and Maximilian Wötzel. On testing minor-freeness in bounded degree graphs with one-sided error. Unpublished manuscript, 2017. Google Scholar
  7. R. Levi, G. Moshkovitz, D. Ron, R. Rubinfeld, and A. Shapira. Constructing near spanning trees with few local inspections. Random Structures &Algorithms, 50:n/a-n/a, 2016. URL: http://dx.doi.org/10.1002/rsa.20652.
  8. 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
  9. 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
  10. R. Levi, D. Ron, and R. Rubinfeld. Local algorithms for sparse spanning graphs. CoRR, abs/1402.3609, 2014. URL: http://arxiv.org/abs/1402.3609.
  11. R. Levi, R. Rubinfeld, and A. Yodpinyanee. Local computation algorithms for graphs of non-constant degrees. Algorithmica, pages 1-24, 2016. Google Scholar
  12. 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
  13. 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
  14. D. Peleg and A. A. Schäffer. Graph spanners. Journal of Graph Theory, 13:99-116, 1989. Google Scholar
  15. D. Peleg and J. D. Ullman. An optimal synchronizer for the hypercube. SIAM Journal on Computing, 18:229-243, 1989. Google Scholar
  16. L. S. Ram and E. Vicari. Distributed small connected spanning subgraph: Breaking the diameter bound. Technical report, ETH Zürich, 2011. Google Scholar
  17. 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
  18. Ronitt Rubinfeld. Can we locally compute sparse connected subgraphs? In Computer Science - Theory and Applications - 12th International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8-12, 2017, Proceedings, pages 38-47, 2017. URL: http://dx.doi.org/10.1007/978-3-319-58747-9_6.