Skip to main content
Springer Nature Link
Log in

Lightweight Near-Additive Spanners

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14760))

Included in the following conference series:

  • 173 Accesses

Abstract

An \((\alpha ,\beta )\)-spanner of a weighted graph \(G=(V,E)\), is a subgraph H such that for every \(u,v\in V\), \(d_G(u,v) \le d_H(u,v)\le \alpha \cdot d_G(u,v)+\beta \). The main parameters of interest for spanners are their size (number of edges) and their lightness (the ratio between the total weight of H to the weight of a minimum spanning tree).

In this paper we focus on near-additive spanners, where \(\alpha =1+\varepsilon \) for arbitrarily small \(\varepsilon >0\). We show the first construction of light spanners in this setting. Specifically, for any integer parameter \(k\ge 1\), we obtain an \((1+\varepsilon ,O(k/\varepsilon )^k\cdot W(\cdot ,\cdot ))\)-spanner with lightness \(\widetilde{O}(n^{1/k})\) (where \(W(\cdot ,\cdot )\) indicates for every pair \(u, v \in V\) the heaviest edge in some shortest path between u, v). In addition, we can also bound the number of edges in our spanner by \(O(kn^{1+3/k})\).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The notation \(O_{\varepsilon }(\cdot )\) hides \((\frac{1}{\varepsilon })^{O(1)}\) factors.

  2. 2.

    The notation \(\Omega _{k}(\cdot )\) hides \(k^{O(1)}\) factors.

  3. 3.

    The notation \(\widetilde{O}(f(n))\) hides polylogarithmic factors in n.

  4. 4.

    A prioritized spanner receives an arbitrary ranking of the points, and should obtain better stretch for high ranking points.

  5. 5.

    A reliable spanner is susceptible to massive vertex failures \(B\subseteq V\), and still provide meaningful guarantees for all vertex pairs in \(V\setminus B^+\), where \(B^+\) is only slightly larger than B.

References

  1. Abboud, A., Bodwin, G.: The 4/3 additive spanner exponent is tight. J. ACM 64(4), 28:1–28:20 (2017). https://doi.org/10.1145/3088511

  2. Abboud, A., Bodwin, G., Pettie, S.: A hierarchy of lower bounds for sublinear additive spanners. In: Klein, P.N. (ed.) Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, 16–19 January, pp. 568–576. SIAM (2017). https://doi.org/10.1137/1.9781611974782.36

  3. Ahmed, R., Bodwin, G., Hamm, K., Kobourov, S., Spence, R.: On additive spanners in weighted graphs with local error. In: 47th International Workshop on Graph-Theoretic Concepts in Computer Science (2021)

    Google Scholar 

  4. Ahmed, R., et al.: Graph spanners: a tutorial review. Comput. Sci. Rev. 37, 100–253 (2020). https://doi.org/10.1016/j.cosrev.2020.100253. http://www.sciencedirect.com/science/article/pii/S1574013719302539

  5. Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28, 1167–1181 (1999). https://doi.org/10.1137/S0097539796303421

    Article  MathSciNet  MATH  Google Scholar 

  6. Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discret. Comput. Geom. 9(1), 81–100 (1993). https://doi.org/10.1007/BF02189308

    Article  MathSciNet  MATH  Google Scholar 

  7. Bartal, Y., Filtser, A., Neiman, O.: On notions of distortion and an almost minimum spanning tree with constant average distortion. J. Comput. Syst. Sci. 105, 116–129 (2019). https://doi.org/10.1016/J.JCSS.2019.04.006

    Article  MathSciNet  MATH  Google Scholar 

  8. Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: Additive spanners and (\(\alpha \), \(\beta \))-spanners. ACM Trans. Algorithms (TALG) 7(1), 5 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Bodwin, G.: An alternate proof of near-optimal light spanners. CoRR abs/2305.18647 (2023). https://doi.org/10.48550/arXiv.2305.18647

  10. Bodwin, G., Williams, V.V.: Very sparse additive spanners and emulators. In: Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science, ITCS 2015, pp. 377–382. Association for Computing Machinery, New York, NY, USA (2015). https://doi.org/10.1145/2688073.2688103

  11. Borradaile, G., Le, H., Wulff-Nilsen, C.: Minor-free graphs have light spanners. In: 2017 IEEE 58th Annual Symposium on Foundations of Computer Science, FOCS 2017, pp. 767–778 (2017). https://doi.org/10.1109/FOCS.2017.76

  12. Borradaile, G., Le, H., Wulff-Nilsen, C.: Greedy spanners are optimal in doubling metrics. In: Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, pp. 2371–2379 (2019). https://doi.org/10.1137/1.9781611975482.145

  13. Chandra, B., Das, G., Narasimhan, G., Soares, J.: New sparseness results on graph spanners. In: Proceedings of the Eighth Annual Symposium on Computational Geometry, pp. 192–201. ACM (1992)

    Google Scholar 

  14. Chechik, S.: New additive spanners. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 498–512. Society for Industrial and Applied Mathematics (2013)

    Google Scholar 

  15. Chechik, S., Wulff-Nilsen, C.: Near-optimal light spanners. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 883–892. Society for Industrial and Applied Mathematics (2016)

    Google Scholar 

  16. Elkin, M., Gitlitz, Y., Neiman, O.: Almost shortest paths with near-additive error in weighted graphs. In: Czumaj, A., Xin, Q. (eds.) 18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022, Tórshavn, Faroe Islands, 27–29 June 2022. LIPIcs, vol. 227, pp. 23:1–23:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022). https://doi.org/10.4230/LIPICS.SWAT.2022.23

  17. Elkin, M., Gitlitz, Y., Neiman, O.: Improved weighted additive spanners. Distrib. Comput. 36(3), 385–394 (2023). https://doi.org/10.1007/S00446-022-00433-X

    Article  MathSciNet  MATH  Google Scholar 

  18. Elkin, M., Neiman, O.: Efficient algorithms for constructing very sparse spanners and emulators. ACM Trans. Algorithms 15(1), 4:1–4:29 (2019). https://doi.org/10.1145/3274651

  19. Elkin, M., Neiman, O., Solomon, S.: Light spanners. In: International Colloquium on Automata, Languages, and Programming, pp. 442–452. Springer (2014)

    Google Scholar 

  20. Elkin, M., Peleg, D.: \((1+\varepsilon ,\beta )\)-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004). https://doi.org/10.1137/S0097539701393384

  21. Erdős, P.: Extremal problems in graph theory. In: Proceedings of the Symposium on Theory of Graphs and its Applications, pp. 29–36 (1963)

    Google Scholar 

  22. Filtser, A., Gitlitz, Y., Neiman, O.: Light, reliable spanners. In: SOCG 2024 (2024, to apear)

    Google Scholar 

  23. Filtser, A., Neiman, O.: Light spanners for high dimensional norms via stochastic decompositions. Algorithmica 84(10), 2987–3007 (2022). https://doi.org/10.1007/S00453-022-00994-0

    Article  MathSciNet  MATH  Google Scholar 

  24. Filtser, A., Solomon, S.: The greedy spanner is existentially optimal. SIAM J. Comput. 49(2), 429–447 (2020). https://doi.org/10.1137/18M1210678, preliminary version published in PODC 2016

  25. Gottlieb, L.: A light metric spanner. In: IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17–20 October 2015, pp. 759–772 (2015). https://doi.org/10.1109/FOCS.2015.52

  26. Khuller, S., Raghavachari, B., Young, N.: Balancing minimum spanning trees and shortest-path trees. Algorithmica 14(4), 305–321 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Le, H., Solomon, S.: A unified framework for light spanners. In: Saha, B., Servedio, R.A. (eds.) Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, 20–23 June 2023, pp. 295–308. ACM (2023). https://doi.org/10.1145/3564246.3585185

  28. Neiman, O., Shabat, I.: A unified framework for hopsets. In: Chechik, S., Navarro, G., Rotenberg, E., Herman, G. (eds.) 30th Annual European Symposium on Algorithms, ESA 2022, Berlin/Potsdam, Germany, 5–9 September 2022. LIPIcs, vol. 244, pp. 81:1–81:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022). https://doi.org/10.4230/LIPICS.ESA.2022.81

  29. Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  30. Pettie, S.: Low distortion spanners. ACM Trans. Algorithms 6(1), 7:1–7:22 (2009). https://doi.org/10.1145/1644015.1644022

  31. Salowe, J.S.: Construction of multidimensional spanner graphs, with applications to minimum spanning trees. In: Proceedings of the Seventh Annual Symposium on Computational Geometry, SCG 1991, pp. 256–261. Association for Computing Machinery, New York, NY, USA (1991). https://doi.org/10.1145/109648.109677

  32. Tan, Z., Zhang, T.: Almost-optimal sublinear additive spanners. In: Saha, B., Servedio, R.A. (eds.) Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, 20–23 June 2023, pp. 281–294. ACM (2023). https://doi.org/10.1145/3564246.3585125

  33. Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005). https://doi.org/10.1145/1044731.1044732

    Article  MathSciNet  MATH  Google Scholar 

  34. Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: SODA 2006, pp. 802–809 (2006). https://doi.org/10.1145/1109557.1109645

  35. Vaidya, P.M.: A sparse graph almost as good as the complete graph on points in K dimensions. Discret. Comput. Geom. 6(3), 369–381 (1991). https://doi.org/10.1007/BF02574695

    Article  MathSciNet  MATH  Google Scholar 

  36. Woodruff, D.: Additive spanners in nearly quadratic time. In: International Colloquium on Automata, Languages, and Programming, pp. 463–474. Springer (2010)

    Google Scholar 

Download references

Acknowledgments

This study was funded by Israel Science Foundation (grant No. 970/21).

Author information

Authors and Affiliations

  1. Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Yuval Gitlitz, Ofer Neiman & Richard Spence

Authors
  1. Yuval Gitlitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ofer Neiman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Richard Spence
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuval Gitlitz .

Editor information

Editors and Affiliations

  1. Masaryk University, Brno, Czech Republic

    Daniel Kráľ

  2. University of Primorska, Koper, Slovenia

    Martin Milanič

Rights and permissions

Reprints and permissions

Copyright information

© 2025 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Gitlitz, Y., Neiman, O., Spence, R. (2025). Lightweight Near-Additive Spanners. In: Kráľ, D., Milanič, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2024. Lecture Notes in Computer Science, vol 14760. Springer, Cham. https://doi.org/10.1007/978-3-031-75409-8_17

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-75409-8_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-75408-1

  • Online ISBN: 978-3-031-75409-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics