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Distributed Algorithms for Low Stretch Spanning Trees

Authors Ruben Becker, Yuval Emek, Mohsen Ghaffari, Christoph Lenzen

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Author Details

Ruben Becker
  • Gran Sasso Science Institute, L'Aquila, Italy
Yuval Emek
  • Technion - Israel Institute of Technology, Haifa, Israel
Mohsen Ghaffari
  • ETH Zurich, Switzerland
Christoph Lenzen
  • MPI for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Becker, Ruben: This work has been partially supported by the Italian MIUR PRIN 2017 Project ALGADIMAR "Algorithms, Games, and Digital Markets".
  • Emek, Yuval: This work has been supported in part by an Israeli Science Foundation grant number 1016/17.

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Ruben Becker, Yuval Emek, Mohsen Ghaffari, and Christoph Lenzen. Distributed Algorithms for Low Stretch Spanning Trees. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.DISC.2019.4

Abstract

Given an undirected graph with integer edge lengths, we study the problem of approximating the distances in the graph by a spanning tree based on the notion of stretch. Our main contribution is a distributed algorithm in the CONGEST model of computation that constructs a random spanning tree with the guarantee that the expected stretch of every edge is O(log^{3} n), where n is the number of nodes in the graph. If the graph is unweighted, then this algorithm can be implemented to run in O(D) rounds, where D is the hop-diameter of the graph, thus being asymptotically optimal. In the weighted case, the run-time of our algorithm matches the currently best known bound for exact distance computations, i.e., O~ (min{sqrt{n D}, sqrt{n} D^{1 / 4} + n^{3 / 5} + D}). We stress that this is the first distributed construction of spanning trees leading to poly-logarithmic expected stretch with non-trivial running time.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Distributed algorithms
Keywords
  • distributed graph algorithms
  • low-stretch spanning trees
  • CONGEST model
  • ball decomposition
  • star decomposition

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References

  1. Ittai Abraham, Yair Bartal, and Ofer Neiman. Nearly Tight Low Stretch Spanning Trees. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 781-790, 2008. Google Scholar
  2. Ittai Abraham and Ofer Neiman. Using petal-decompositions to build a low stretch spanning tree. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC, pages 395-406, 2012. Google Scholar
  3. Noga Alon, Richard M. Karp, David Peleg, and Douglas B. West. A Graph-Theoretic Game and Its Application to the k-Server Problem. SIAM J. Comput., 24(1):78-100, 1995. Google Scholar
  4. Reid Andersen and Uriel Feige. Interchanging distance and capacity in probabilistic mappings. CoRR, abs/0907.3631, 2009. URL: http://arxiv.org/abs/0907.3631.
  5. Yair Bartal. Probabilistic Approximations of Metric Spaces and Its Algorithmic Applications. In 37th Annual Symposium on Foundations of Computer Science, FOCS, pages 184-193, 1996. Google Scholar
  6. Yair Bartal. On Approximating Arbitrary Metrices by Tree Metrics. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, STOC, pages 161-168, 1998. Google Scholar
  7. Yair Bartal. Graph Decomposition Lemmas and Their Role in Metric Embedding Methods. In Algorithms - ESA 2004, 12th Annual European Symposium, pages 89-97, 2004. Google Scholar
  8. Moses Charikar, Chandra Chekuri, Ashish Goel, Sudipto Guha, and Serge A. Plotkin. Approximating a Finite Metric by a Small Number of Tree Metrics. In 39th Annual Symposium on Foundations of Computer Science, FOCS, pages 379-388, 1998. Google Scholar
  9. Paul Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, and Shang-Hua Teng. Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC, pages 273-282, 2011. Google Scholar
  10. Michael B. Cohen, Brittany Terese Fasy, Gary L. Miller, Amir Nayyeri, Richard Peng, and Noel Walkington. Solving 1-Laplacians in Nearly Linear Time: Collapsing and Expanding a Topological Ball. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 204-216, 2014. Google Scholar
  11. Michael B. Cohen, Rasmus Kyng, Gary L. Miller, Jakub W. Pachocki, Richard Peng, Anup B. Rao, and Shen Chen Xu. Solving SDD linear systems in nearly mlog^1/2n time. In Symposium on Theory of Computing, STOC, pages 343-352, 2014. Google Scholar
  12. Michael Elkin, Yuval Emek, Daniel A. Spielman, and Shang-Hua Teng. Lower-Stretch Spanning Trees. SIAM J. Comput., 38(2):608-628, 2008. Google Scholar
  13. Matthias Englert and Harald Räcke. Oblivious Routing for the Lp-norm. In 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 32-40, 2009. Google Scholar
  14. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485-497, 2004. Google Scholar
  15. Sebastian Forster and Danupon Nanongkai. A Faster Distributed Single-Source Shortest Paths Algorithm. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS, pages 686-697, 2018. Google Scholar
  16. Stephan Friedrichs and Christoph Lenzen. Parallel Metric Tree Embedding Based on an Algebraic View on Moore-Bellman-Ford. J. ACM, 65(6):43:1-43:55, 2018. Google Scholar
  17. Juan A Garay, Shay Kutten, and David Peleg. A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM Journal on Computing, 27(1):302-316, 1998. Google Scholar
  18. Mohsen Ghaffari, Andreas Karrenbauer, Fabian Kuhn, Christoph Lenzen, and Boaz Patt-Shamir. Near-Optimal Distributed Maximum Flow: Extended Abstract. In Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC, pages 81-90, 2015. Google Scholar
  19. Mohsen Ghaffari and Christoph Lenzen. Near-Optimal Distributed Tree Embedding. In Distributed Computing - 28th International Symposium, DISC, pages 197-211, 2014. Google Scholar
  20. Jonathan A. Kelner, Yin Tat Lee, Lorenzo Orecchia, and Aaron Sidford. An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 217-226, 2014. Google Scholar
  21. Jonathan A. Kelner, Lorenzo Orecchia, Aaron Sidford, and Zeyuan Allen Zhu. A simple, combinatorial algorithm for solving SDD systems in nearly-linear time. In Symposium on Theory of Computing Conference, STOC, pages 911-920, 2013. Google Scholar
  22. Maleq Khan, Fabian Kuhn, Dahlia Malkhi, Gopal Pandurangan, and Kunal Talwar. Efficient distributed approximation algorithms via probabilistic tree embeddings. Distributed Computing, 25(3):189-205, 2012. Google Scholar
  23. Ioannis Koutis, Gary L. Miller, and Richard Peng. A Nearly-m log n Time Solver for SDD Linear Systems. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, October 22-25, 2011, pages 590-598, 2011. Google Scholar
  24. Shay Kutten and David Peleg. Fast Distributed Construction of Smallk-Dominating Sets and Applications. Journal of Algorithms, 28(1):40-66, 1998. Google Scholar
  25. Yin Tat Lee, Satish Rao, and Nikhil Srivastava. A new approach to computing maximum flows using electrical flows. In Symposium on Theory of Computing Conference, STOC, pages 755-764, 2013. Google Scholar
  26. Aleksander Madry. Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 245-254, 2010. Google Scholar
  27. Gary L. Miller, Richard Peng, and Shen Chen Xu. Parallel graph decompositions using random shifts. In 25th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '13, Montreal, QC, Canada - July 23 - 25, 2013, pages 196-203, 2013. Google Scholar
  28. D. Peleg. Distributed Computing: A Locality-Sensitive Approach. Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, 2000. Google Scholar
  29. Harald Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing, STOC, pages 255-264, 2008. Google Scholar
  30. Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. Distributed Verification and Hardness of Distributed Approximation. SIAM J. Comput., 41(5):1235-1265, 2012. Google Scholar
  31. Jonah Sherman. Nearly Maximum Flows in Nearly Linear Time. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pages 263-269, 2013. Google Scholar
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