Skip to main content
SpringerLink
Log in

Distributed Construction of Purely Additive Spanners

  • Conference paper
  • First Online:
Distributed Computing (DISC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9888))

Included in the following conference series:

Abstract

This paper studies the complexity of distributed construction of purely additive spanners in the CONGEST model. We describe algorithms for building such spanners in several cases. Because of the need to simultaneously make decisions at far apart locations, the algorithms use additional mechanisms compared to their sequential counterparts.

We complement our algorithms with a lower bound on the number of rounds required for computing pairwise spanners. The standard reductions from set-disjointness and equality seem unsuitable for this task because no specific edge needs to be removed from the graph. Instead, to obtain our lower bound, we define a new communication complexity problem that reduces to computing a sparse spanner, and prove a lower bound on its communication complexity using information theory. This technique significantly extends the current toolbox used for obtaining lower bounds for the CONGEST model, and we believe it may find additional applications.

Keren Censor-Hillel and Ami Paz were Supported by ISF individual research grant 1696/14. Part of this work was done while Ami Paz was visiting TIFR, Mumbai.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The girth of a graph is the length of the shortest simple cycle in it.

  2. 2.

    In fact, our lower bound holds even if all nodes in pairs in \({\mathcal {P}}\) know all of \({\mathcal {P}}\).

References

  1. Abboud, A., Bodwin, G.: The 4/3 additive spanner exponent is tight. In: ACM SIGACT Symposium on Theory of Computing, STOC (2016)

    Google Scholar 

  2. Abboud, A., Bodwin, G.: Error amplification for pairwise spanner lower bounds. In: Annual ACM-SIAM Symposium on Discrete Algorithms, SODA (2016)

    Google Scholar 

  3. Aingworth, D., Chekuri, C., Indyk, P., Motwani, R.: Fast estimation of diameter and shortest paths (without matrix multiplication). SIAM J. Comput. 28(4), 1167–1181 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Althöfer, I., Das, G., Dobkin, D.P., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geom. 9, 81–100 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baswana, S.: Streaming algorithm for graph spanners - single pass and constant processing time per edge. Inf. Process. Lett. 106(3), 110–114 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Baswana, S., Khurana, S., Sarkar, S.: Fully dynamic randomized algorithms for graph spanners. ACM Trans. Algorithms 8(4), 35 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Baswana, S., Sarkar, S.: Fully dynamic algorithm for graph spanners with poly-logarithmic update time. In: ACM-SIAM Symposium on Discrete Algorithms, SODA (2008)

    Google Scholar 

  9. Baswana, S., Sen, S.: A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Struct. Algorithm 30(4), 532–563 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bodwin, G., Williams, V.V.: Better distance preservers and additive spanners. In: ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 855–872 (2016)

    Google Scholar 

  11. Bollobás, B., Coppersmith, D., Elkin, M.: Sparse distance preservers and additive spanners. SIAM J. Discrete Math. 19(4), 1029–1055 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Censor-Hillel, K., Ghaffari, M., Kuhn, F.: Distributed connectivity decomposition. In: ACM Symposium on Principles of Distributed Computing, PODC (2014)

    Google Scholar 

  13. Censor-Hillel, K., Haeupler, B., Kelner, J.A., Maymounkov, P.: Global computation in a poorly connected world: fast rumor spreading with no dependence on conductance. In: Symposium on Theory of Computing Conference, STOC, 2012

    Google Scholar 

  14. Chechik, S.: Compact routing schemes with improved stretch. In: ACM Symposium on Principles of Distributed Computing, PODC (2013)

    Google Scholar 

  15. Chechik, S.: New additive spanners. In: ACM-SIAM Symposium on Discrete Algorithms, SODA (2013)

    Google Scholar 

  16. Coppersmith, D., Elkin, M.: Sparse sourcewise and pairwise distance preservers. SIAM J. Discrete Math. 20(2), 463–501 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cygan, M., Grandoni, F., Kavitha, T.: On pairwise spanners. In: Symposium on Theoretical Aspects of Computer Science, STACS (2013)

    Google Scholar 

  18. Das Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. SIAM J. Comput. 41(5), 1235–1265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Derbel, B., Gavoille, C.: Fast deterministic distributed algorithms for sparse spanners. Theor. Comput. Sci. 399(1–2), 83–100 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Derbel, B., Gavoille, C., Peleg, D.: Deterministic distributed construction of linear stretch spanners in polylogarithmic time. In: Pelc, A. (ed.) DISC 2007. LNCS, vol. 4731, pp. 179–192. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  21. Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 273–282 (2008)

    Google Scholar 

  22. Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: Local computation of nearly additive spanners. In: Keidar, I. (ed.) DISC 2009. LNCS, vol. 5805, pp. 176–190. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  23. Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29(5), 1740–1759 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Drucker, A., Kuhn, F., Oshman, R.: On the power of the congested clique model. In: ACM Symposium on Principles of Distributed Computing, PODC (2014)

    Google Scholar 

  25. Dubhashi, D.P., Mei, A., Panconesi, A., Radhakrishnan, J., Srinivasan, A.: Fast distributed algorithms for (weakly) connected dominating sets and linear-size skeletons. J. Comput. Syst. Sci. 71(4), 467–479 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Elkin, M.: Computing almost shortest paths. ACM Trans. Algorithm 1(2), 283–323 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Elkin, M.: A near-optimal distributed fully dynamic algorithm for maintaining sparse spanners. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 185–194 (2007)

    Google Scholar 

  28. Elkin, M.: Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 716–727. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  29. Elkin, M., Peleg, D.: (1+epsilon, beta)-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Elkin, M., Zhang, J.: Efficient algorithms for constructing (1+epsilon, beta)-spanners in the distributed and streaming models. Distrib. Comput. 18(5), 375–385 (2006)

    Article  MATH  Google Scholar 

  31. Frischknecht, S., Holzer, S., Wattenhofer, R.: Networks cannot compute their diameter in sublinear time. In: ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 1150–1162 (2012)

    Google Scholar 

  32. Ghaffari, M., Kuhn, F.: Distributed minimum cut approximation. In: Afek, Y. (ed.) DISC 2013. LNCS, vol. 8205, pp. 1–15. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  33. Holzer, S., Pinsker, N.: Approximation of distances and shortest paths in the broadcast congest clique. CoRR, abs/1412.3445 (2014)

    Google Scholar 

  34. Holzer, S., Wattenhofer, R.: Optimal distributed all pairs shortest paths and applications. In: ACM Symposium on Principles of Distributed Computing, PODC (2012)

    Google Scholar 

  35. Kavitha, T.: New pairwise spanners. In: Symposium on Theoretical Aspects of Computer Science, STACS, pp. 513–526 (2015)

    Google Scholar 

  36. Kavitha, T., Varma, N.M.: Small stretch pairwise spanners. In: International Colloquium on Automata, Languages, and Programming, ICALP (2013)

    Google Scholar 

  37. Knudsen, M.B.T.: Additive spanners: a simple construction. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 277–281. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  38. Lenzen, C., Peleg, D.: Efficient distributed source detection with limited bandwidth. In: ACM Symposium on Principles of Distributed Computing, PODC (2013)

    Google Scholar 

  39. Matousek, J.: Lectures on Discrete Geometry. Springer, New York (2002)

    Book  MATH  Google Scholar 

  40. Parter, M.: Bypassing Erdős’ girth conjecture: hybrid stretch and sourcewise spanners. In: International Colloquium on Automata, Languages and Programming, ICALP (2014)

    Google Scholar 

  41. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2000)

    Book  MATH  Google Scholar 

  42. Peleg, D., Rubinovich, V.: A near-tight lower bound on the time complexity of distributed MST construction. In: Symposium on Foundations of Computer Science, FOCS, pp. 253–261 (1999)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comput. 18(4), 740–747 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  45. Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. J. ACM 36(3), 510–530 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  46. Pettie, S.: Low distortion spanners. ACM Trans. Algorithms 6(1) (2009)

    Google Scholar 

  47. Pettie, S.: Distributed algorithms for ultrasparse spanners and linear size skeletons. Distrib. Comput. 22(3), 147–166 (2010)

    Article  MATH  Google Scholar 

  48. Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 261–272. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  49. Roditty, L., Zwick, U.: On dynamic shortest paths problems. Algorithmica 61(2), 389–401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Thorup, M., Zwick, U.: Compact routing schemes. In: ACM Symposium on Parallel Algorithms and Architectures, SPAA, pp. 1–10 (2001)

    Google Scholar 

  51. Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 113–116 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  52. Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: ACM-SIAM Symposium on Discrete Algorithms, SODA, pp. 802–809 (2006)

    Google Scholar 

  53. Woodruff, D.P.: Additive spanners in nearly quadratic time. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 463–474. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

Download references

Acknowledgements

We thank Merav Parter for a helpful discussion on the lower bound, and the anonymous referees for helpful comments.

Author information

Authors and Affiliations

  1. Department of Computer Science, Technion, Haifa, Israel

    Keren Censor-Hillel & Ami Paz

  2. Tata Institute of Fundamental Research, Mumbai, India

    Telikepalli Kavitha

  3. Department of Mathematics, Technion, Haifa, Israel

    Amir Yehudayoff

Authors
  1. Keren Censor-Hillel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Telikepalli Kavitha
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ami Paz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Amir Yehudayoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ami Paz .

Editor information

Editors and Affiliations

  1. LABRI, Univ of Bordeaux LABRI, Talence Cedex, France

    Cyril Gavoille

  2. CNRS, Universite Bordeaux 1 CNRS, Talence Cedex, France

    David Ilcinkas

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Censor-Hillel, K., Kavitha, T., Paz, A., Yehudayoff, A. (2016). Distributed Construction of Purely Additive Spanners. In: Gavoille, C., Ilcinkas, D. (eds) Distributed Computing. DISC 2016. Lecture Notes in Computer Science(), vol 9888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53426-7_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-53426-7_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-53425-0

  • Online ISBN: 978-3-662-53426-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation