A Unified Framework for Hopsets

Authors Ofer Neiman, Idan Shabat

Thumbnail PDF


  • Filesize: 0.73 MB
  • 13 pages

Document Identifiers

Author Details

Ofer Neiman
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Idan Shabat
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel


We want to acknowledge the review of the anonymous reviewers from SODA 2022. This review drew our attention for some improvements that we applied in the paper.

Cite AsGet BibTex

Ofer Neiman and Idan Shabat. A Unified Framework for Hopsets. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Given an undirected graph G = (V,E), an (α,β)-hopset is a graph H = (V,E'), so that adding its edges to G guarantees every pair has an α-approximate shortest path that has at most β edges (hops), that is, d_G(u,v) ≤ d_{G∪H}^(β)(u,v) ≤ α⋅ d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter α. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^ε),O(k^{1-ε}))-hopset of size Õ(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ε < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (α,β)-hopset of size O(n^{1+1/k}) must have α⋅β ≥ Ω(k).

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
  • Graph Algorithms
  • Shortest Paths
  • Hopsets


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


  1. Amir Abboud, Greg Bodwin, and Seth Pettie. A hierarchy of lower bounds for sublinear additive spanners. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 568-576, 2017. URL: https://doi.org/10.1137/1.9781611974782.36.
  2. Alexandr Andoni, Clifford Stein, and Peilin Zhong. Parallel approximate undirected shortest paths via low hop emulators. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, page 322?335, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3357713.3384321.
  3. Uri Ben-Levy and Merav Parter. New (a, ?) spanners and hopsets. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '20, page 1695?1714, USA, 2020. Society for Industrial and Applied Mathematics. Google Scholar
  4. Keren Censor-Hillel, Michal Dory, Janne H. Korhonen, and Dean Leitersdorf. Fast approximate shortest paths in the congested clique. In Peter Robinson and Faith Ellen, editors, Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29 - August 2, 2019, pages 74-83. ACM, 2019. URL: https://doi.org/10.1145/3293611.3331633.
  5. Edith Cohen. Polylog-time and near-linear work approximation scheme for undirected shortest paths. J. ACM, 47(1):132-166, 2000. URL: https://doi.org/10.1145/331605.331610.
  6. Michal Dory and Merav Parter. Exponentially faster shortest paths in the congested clique. In Proceedings of the 39th Symposium on Principles of Distributed Computing, PODC ?20, page 59?68, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3382734.3405711.
  7. Michael Elkin, Yuval Gitlitz, and Ofer Neiman. Almost shortest paths with near-additive error in weighted graphs. CoRR, abs/1907.11422, 2019. URL: http://arxiv.org/abs/1907.11422.
  8. Michael Elkin and Ofer Neiman. Hopsets with constant hopbound, and applications to approximate shortest paths. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 128-137, 2016. URL: https://doi.org/10.1109/FOCS.2016.22.
  9. Michael Elkin and Ofer Neiman. On efficient distributed construction of near optimal routing schemes. Distributed Comput., 31(2):119-137, 2018. URL: https://doi.org/10.1007/s00446-017-0304-4.
  10. Michael Elkin and Ofer Neiman. Linear-size hopsets with small hopbound, and constant-hopbound hopsets in RNC. In The 31st ACM on Symposium on Parallelism in Algorithms and Architectures, SPAA 2019, Phoenix, AZ, USA, June 22-24, 2019., pages 333-341, 2019. URL: https://doi.org/10.1145/3323165.3323177.
  11. Michael Elkin and Ofer Neiman. Centralized and parallel multi-source shortest paths via hopsets and fast matrix multiplication. CoRR, abs/2004.07572, 2020. URL: http://arxiv.org/abs/2004.07572.
  12. Michael Elkin and David Peleg. (1+epsilon, beta)-spanner constructions for general graphs. SIAM J. Comput., 33(3):608-631, 2004. URL: https://doi.org/10.1137/S0097539701393384.
  13. Stephan Friedrichs and Christoph Lenzen. Parallel metric tree embedding based on an algebraic view on moore-bellman-ford. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '16, pages 455-466, New York, NY, USA, 2016. ACM. URL: https://doi.org/10.1145/2935764.2935777.
  14. Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. Decremental single-source shortest paths on undirected graphs in near-linear total update time. J. ACM, 65(6):36:1-36:40, 2018. URL: https://doi.org/10.1145/3218657.
  15. Shang-En Huang and Seth Pettie. Thorup-zwick emulators are universally optimal hopsets. Information Processing Letters, 142, April 2017. URL: https://doi.org/10.1016/j.ipl.2018.10.001.
  16. Philip N. Klein and Sairam Subramanian. A randomized parallel algorithm for single-source shortest paths. J. Algorithms, 25(2):205-220, 1997. URL: https://doi.org/10.1006/jagm.1997.0888.
  17. Jakub Lacki and Yasamin Nazari. Near-optimal decremental approximate multi-source shortest paths. CoRR, abs/2009.08416, 2020. URL: http://arxiv.org/abs/2009.08416.
  18. Christoph Lenzen, Boaz Patt-Shamir, and David Peleg. Distributed distance computation and routing with small messages. Distributed Comput., 32(2):133-157, 2019. URL: https://doi.org/10.1007/s00446-018-0326-6.
  19. Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. Google Scholar
  20. Gary L. Miller, Richard Peng, Adrian Vladu, and Shen Chen Xu. Improved parallel algorithms for spanners and hopsets. In Proceedings of the 27th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '15, pages 192-201, New York, NY, USA, 2015. ACM. URL: https://doi.org/10.1145/2755573.2755574.
  21. M. Thorup and U. Zwick. Approximate distance oracles. In Proc. of the 33rd ACM Symp. on Theory of Computing, pages 183-192, 2001. Google Scholar
  22. M. Thorup and U. Zwick. Spanners and emulators with sublinear distance errors. In Proc. of Symp. on Discr. Algorithms, pages 802-809, 2006. Google Scholar