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Light Edge Fault Tolerant Graph Spanners

Authors Greg Bodwin , Michael Dinitz , Ama Koranteng, Lily Wang

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  • Theory of computation → Sparsification and spanners
Keywords
  • Fault Tolerant Spanners
  • Light Spanners

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Abstract

There has recently been significant interest in fault tolerant spanners, which are spanners that still maintain their stretch guarantees after some nodes or edges fail. This work has culminated in an almost complete understanding of the three-way tradeoff between stretch, sparsity, and number of faults tolerated. However, despite some progress in metric settings, there have been no results to date on the tradeoff in general graphs between stretch, lightness, and number of faults tolerated.
We initiate the study of light edge fault tolerant (EFT) graph spanners, obtaining the first such results. First, we observe that lightness can be unbounded if we use the traditional definition (normalizing by the MST). We then argue that a natural definition of fault-tolerant lightness is to instead normalize by a min-weight fault tolerant connectivity preserver; essentially, a fault-tolerant version of the MST. However, even with this, we show that it is still not generally possible to construct f-EFT spanners whose weight compares reasonably to the weight of a min-weight f-EFT connectivity preserver.
In light of this lower bound, it is natural to then consider bicriteria notions of lightness, where we compare the weight of an f-EFT spanner to a min-weight (f' > f)-EFT connectivity preserver. The most interesting question is to determine the minimum value of f' that allows for reasonable lightness upper bounds. Our main result is a precise answer to this question: f' = 2f. In particular, we show that the lightness can be untenably large (roughly n/k for a k-spanner) if one normalizes by the min-weight (2f-1)-EFT connectivity preserver. But if one normalizes by the min-weight 2f-EFT connectivity preserver, then we show that the lightness is bounded by just O(f^{1/2}) times the non-fault tolerant lightness (roughly n^{1/k} for a (1+ε)(2k-1)-spanner).

Cite As Get BibTex

Greg Bodwin, Michael Dinitz, Ama Koranteng, and Lily Wang. Light Edge Fault Tolerant Graph Spanners. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.32

Author Details

Greg Bodwin
  • University of Michigan, Ann Arbor, MI, USA
Michael Dinitz
  • Johns Hopkins University, Baltimore, MD, USA
Ama Koranteng
  • Johns Hopkins University, Baltimore, MD, USA
Lily Wang
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Bodwin, Greg: Supported by NSF:AF 2153680.
  • Dinitz, Michael: Supported in part by NSF award 2228995.
  • Koranteng, Ama: Supported in part by an NSF Graduate Research Fellowship.
  • Wang, Lily: Supported by NSF:AF 2153680.

References

  1. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9(1):81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  2. Baruch Awerbuch, Alan Baratz, and David Peleg. Efficient broadcast and light-weight spanners. Unpublished manuscript, November, 1991. Google Scholar
  3. Baruch Awerbuch and David Peleg. Network synchronization with polylogarithmic overhead. In Foundations of Computer Science, 1990. Proceedings., 31st Annual Symposium on, pages 514-522. IEEE, 1990. URL: https://doi.org/10.1109/FSCS.1990.89572.
  4. Georg Baier, Thomas Erlebach, Alexander Hall, Ekkehard Köhler, Petr Kolman, Ondřej Pangrác, Heiko Schilling, and Martin Skutella. Length-bounded cuts and flows. ACM Transactions on Algorithms (TALG), 7(1):1-27, 2010. URL: https://doi.org/10.1145/1868237.1868241.
  5. Greg Bodwin. An alternate proof of near-optimal light spanners. TheoretiCS, 4, 2025. URL: https://doi.org/10.46298/THEORETICS.25.2.
  6. Greg Bodwin, Michael Dinitz, Ama Koranteng, and Lily Wang. Light edge fault tolerant graph spanners, 2025. URL: https://doi.org/10.48550/arXiv.2502.10890.
  7. Greg Bodwin, Michael Dinitz, and Yasamin Nazari. Vertex Fault-Tolerant Emulators. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 25:1-25:22, 2022. URL: https://doi.org/10.4230/LIPICS.ITCS.2022.25.
  8. Greg Bodwin, Michael Dinitz, and Yasamin Nazari. Epic Fail: Emulators Can Tolerate Polynomially Many Edge Faults for Free. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023), volume 251, pages 20:1-20:22, 2023. URL: https://doi.org/10.4230/LIPICS.ITCS.2023.20.
  9. Greg Bodwin, Michael Dinitz, Merav Parter, and Virginia Vassilevska Williams. Optimal vertex fault tolerant spanners (for fixed stretch). In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1884-1900. Society for Industrial and Applied Mathematics, 2018. URL: https://doi.org/10.1137/1.9781611975031.123.
  10. Greg Bodwin, Michael Dinitz, and Caleb Robelle. Optimal vertex fault-tolerant spanners in polynomial time. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2924-2938. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.174.
  11. Greg Bodwin, Michael Dinitz, and Caleb Robelle. Partially optimal edge fault-tolerant spanners. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3272-3286. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.129.
  12. Greg Bodwin and Jeremy Flics. A lower bound for light spanners in general graphs. In Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4327-4337. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.146.
  13. Greg Bodwin, Bernhard Haeupler, and Merav Parter. Fault-tolerant spanners against bounded-degree edge failures: Linearly more faults, almost for free. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2609-2642. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.93.
  14. Greg Bodwin and Shyamal Patel. A trivial yet optimal solution to vertex fault tolerant spanners. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, pages 541-543, 2019. URL: https://doi.org/10.1145/3293611.3331588.
  15. Prosenjit Bose, Vida Dujmovic, Pat Morin, and Michiel Smid. Robust geometric spanners. SIAM Journal on Computing, 42(4):1720-1736, 2013. URL: https://doi.org/10.1137/120874473.
  16. Kevin Buchin, Sariel Har-Peled, and Dániel Oláh. A spanner for the day after. Discrete & Computational Geometry, 64(4):1167-1191, 2020. URL: https://doi.org/10.1007/S00454-020-00228-6.
  17. T-H Hubert Chan, Mingfei Li, and Li Ning. Sparse fault-tolerant spanners for doubling metrics with bounded hop-diameter or degree. Algorithmica, 71(1):53-65, 2015. URL: https://doi.org/10.1007/S00453-013-9779-Y.
  18. T-H Hubert Chan, Mingfei Li, Li Ning, and Shay Solomon. New doubling spanners: Better and simpler. SIAM Journal on Computing, 44(1):37-53, 2015. URL: https://doi.org/10.1137/130930984.
  19. Barun Chandra, Gautam Das, Giri Narasimhan, and José Soares. New sparseness results on graph spanners. In Proceedings of the eighth annual symposium on Computational geometry, pages 192-201. ACM, 1992. URL: https://doi.org/10.1145/142675.142717.
  20. Shiri Chechik, Michael Langberg, David Peleg, and Liam Roditty. Fault tolerant spanners for general graphs. SIAM Journal on Computing, 39(7):3403-3423, 2010. URL: https://doi.org/10.1137/090758039.
  21. Shiri Chechik and Christian Wulff-Nilsen. Near-optimal light spanners. In Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, pages 883-892. Society for Industrial and Applied Mathematics, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH63.
  22. Chandra Chekuri and F Bruce Shepherd. Approximate integer decompositions for undirected network design problems. SIAM Journal on Discrete Mathematics, 23(1):163-177, 2009. URL: https://doi.org/10.1137/040617339.
  23. Joseph Cheriyan and Ramakrishna Thurimella. Approximating minimum-size k-connected spanning subgraphs via matching. SIAM Journal on Computing, 30(2):528-560, 2000. URL: https://doi.org/10.1137/S009753979833920X.
  24. Artur Czumaj and Hairong Zhao. Fault-tolerant geometric spanners. Discrete & Computational Geometry, 32(2):207-230, 2004. URL: https://doi.org/10.1007/S00454-004-1121-7.
  25. Matt DeVos, Jessica McDonald, and Irene Pivotto. Packing steiner trees. Journal of Combinatorial Theory, Series B, 119:178-213, 2016. URL: https://doi.org/10.1016/j.jctb.2016.02.002.
  26. Michael Dinitz, Ama Koranteng, and Guy Kortsarz. Relative Survivable Network Design. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022), pages 41:1-41:19, 2022. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2022.41.
  27. Michael Dinitz, Ama Koranteng, Guy Kortsarz, and Zeev Nutov. Improved approximations for relative survivable network design. CoRR, abs/2304.06656, 2023. URL: https://doi.org/10.48550/arXiv.2304.06656.
  28. Michael Dinitz and Robert Krauthgamer. Fault-tolerant spanners: better and simpler. In Proceedings of the 30th annual ACM SIGACT-SIGOPS symposium on Principles of distributed computing, pages 169-178. ACM, 2011. URL: https://doi.org/10.1145/1993806.1993830.
  29. Michael Dinitz and Caleb Robelle. Efficient and simple algorithms for fault-tolerant spanners. In Proceedings of the 39th Symposium on Principles of Distributed Computing, pages 493-500, 2020. URL: https://doi.org/10.1145/3382734.3405735.
  30. Michael Elkin, Yuval Emek, Daniel A Spielman, and Shang-Hua Teng. Lower-stretch spanning trees. SIAM Journal on Computing, 38(2):608-628, 2008. URL: https://doi.org/10.1137/050641661.
  31. Michael Elkin, Ofer Neiman, and Shay Solomon. Light spanners. In International Colloquium on Automata, Languages, and Programming, pages 442-452. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_37.
  32. Paul Erdős. Extremal problems in graph theory. In Proceedings of the Symposium on Theory of Graphs and its Applications, page 2936, 1963. Google Scholar
  33. Harold N. Gabow and Suzanne R. Gallagher. Iterated rounding algorithms for the smallest k-edge connected spanning subgraph. SIAM Journal on Computing, 41(1):61-103, 2012. URL: https://doi.org/10.1137/080732572.
  34. Harold N. Gabow, Michel X. Goemans, Éva Tardos, and David P. Williamson. Approximating the smallest k-edge connected spanning subgraph by lp-rounding. Networks, 53(4):345-357, 2009. URL: https://doi.org/10.1002/NET.20289.
  35. Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Combinatorica, 21(1):39-60, 2001. URL: https://doi.org/10.1007/s004930170004.
  36. Matthias Kriesell. Edge-disjoint trees containing some given vertices in a graph. Journal of Combinatorial Theory, Series B, 88(1):53-65, 2003. URL: https://doi.org/10.1016/S0095-8956(02)00013-8.
  37. Lap Chi Lau. Packing steiner forests. In International Conference on Integer Programming and Combinatorial Optimization, pages 362-376. Springer, 2005. URL: https://doi.org/10.1007/11496915_27.
  38. Hung Le. Recent Progress on Euclidean (and Related) Spanners , 2024. Workshop on Sublinear Graph Simplification, Simons Institute for the Theory of Computing. URL: https://simons.berkeley.edu/talks/hung-le-university-massachusetts-amherst-2024-08-01.
  39. Hung Le and Shay Solomon. A unified framework for light spanners. In Proceedings of the 55th Annual ACM SIGACT Symposium on Theory of Computing (STOC). ACM, 2023. Google Scholar
  40. Hung Le, Shay Solomon, and Cuong Than. Optimal fault-tolerant spanners in euclidean and doubling metrics: Breaking the Ω (log n) lightness barrier. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 77-97. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00013.
  41. Christos Levcopoulos, Giri Narasimhan, and Michiel Smid. Efficient algorithms for constructing fault-tolerant geometric spanners. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pages 186-195. ACM, 1998. URL: https://doi.org/10.1145/276698.276734.
  42. Tamás Lukovszki. New results on fault tolerant geometric spanners. In Workshop on Algorithms and Data Structures, pages 193-204. Springer, 1999. URL: https://doi.org/10.1007/3-540-48447-7_20.
  43. Hiroshi Nagamochi and Toshihide Ibaraki. A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica, 7(5&6):583-596, 1992. URL: https://doi.org/10.1007/BF01758778.
  44. Hiroshi Nagamochi and Toshihide Ibaraki. Algorithmic Aspects of Graph Connectivity, volume 123 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2008. URL: https://doi.org/10.1017/CBO9780511721649.
  45. Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, 2007. URL: https://doi.org/10.1017/CBO9780511546884.
  46. C. St.J. A. Nash-Williams. Edge-Disjoint Spanning Trees of Finite Graphs. Journal of the London Mathematical Society, s1-36(1):445-450, January 1961. URL: https://doi.org/10.1112/jlms/s1-36.1.445.
  47. Merav Parter. Nearly optimal vertex fault-tolerant spanners in optimal time: Sequential, distributed and parallel. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing (STOC). ACM, 2022. Google Scholar
  48. David Peleg. Distributed computing: a locality-sensitive approach. SIAM, 2000. Google Scholar
  49. David Peleg and Alejandro A. Schäffer. Graph spanners. J. Graph Theory, 13(1):99-116, 1989. URL: https://doi.org/10.1002/JGT.3190130114.
  50. David Peleg and Jeffrey Ullman. An optimal synchronizer for the hypercube. SIAM Journal on Computing (SICOMP), 18(4):740-747, 1989. URL: https://doi.org/10.1137/0218050.
  51. David Peleg and Eli Upfal. A trade-off between space and efficiency for routing tables. Journal of the ACM (JACM), 36(3):510-530, 1989. URL: https://doi.org/10.1145/65950.65953.
  52. Asaf Petruschka, Shay Sapir, and Elad Tzalik. Color Fault-Tolerant Spanners. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024), pages 88:1-88:17, 2024. URL: https://doi.org/10.4230/LIPICS.ITCS.2024.88.
  53. Shay Solomon. From hierarchical partitions to hierarchical covers: Optimal fault-tolerant spanners for doubling metrics. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 363-372. ACM, 2014. URL: https://doi.org/10.1145/2591796.2591864.
  54. Mikkel Thorup and Uri Zwick. Approximate distance oracles. Journal of the ACM (JACM), 52(1):1-24, 2005. URL: https://doi.org/10.1145/1044731.1044732.
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