Advertisement

Distributed Computing

, Volume 31, Issue 3, pp 223–240 | Cite as

Distributed construction of purely additive spanners

  • Keren Censor-Hillel
  • Telikepalli Kavitha
  • Ami Paz
  • Amir Yehudayoff
Article
  • 111 Downloads

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. This technique significantly extends the current toolbox used for obtaining lower bounds for the CONGEST model, and we believe it may find additional applications.

Notes

Acknowledgements

We thank Yossi Azar and Uri Zwick for their suggestion which significantly simplified the lower bound proof, Yuval Dagan and Merav Parter for helpful discussions on the lower bound, and the anonymous referees for valuable comments.

References

  1. 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. 2.
    Abboud, A., Bodwin, G.: Error amplification for pairwise spanner lower bounds. In: 27th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, pp. 841–854 (2016)Google Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Althöfer, I., Das, G., Dobkin, D.P., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete Comput. Geometry 9, 81–100 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baswana, S.: Streaming algorithm for graph spanners—single pass and constant processing time per edge. Inf. Process. Lett. 106(3), 110–114 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baswana, S., Sarkar, S.: Fully dynamic algorithm for graph spanners with poly-logarithmic update time. In: 19th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, pp. 1125–1134 (2008)Google Scholar
  7. 7.
    Baswana, S., Sen, S.: A simple and linear time randomized algorithm for computing sparse spanners in weighted graphs. Random Struct. Algorithms 30(4), 532–563 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baswana, S., Kavitha, T., Mehlhorn, K., Pettie, S.: Additive spanners and (\(\alpha, \beta \))-spanners. ACM Trans. Algorithms 7(1), 5 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Baswana, S., Khurana, S., Sarkar, S.: Fully dynamic randomized algorithms for graph spanners. ACM Trans. Algorithms 8(4), 35 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bodwin, G., Williams, V.V.: Better distance preservers and additive spanners. In: 27th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, pp. 855–872 (2016)Google Scholar
  11. 11.
    Bollobás, B., Coppersmith, D., Elkin, M.: Sparse distance preservers and additive spanners. SIAM J. Discrete Math. 19(4), 1029–1055 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Censor-Hillel, K., Ghaffari, M., Kuhn, F.: Distributed connectivity decomposition. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 156–165 (2014)Google Scholar
  13. 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: 44th Symposium on Theory of Computing Conference, STOC, pp. 961–970 (2012)Google Scholar
  14. 14.
    Censor-Hillel, K., Kavitha, T., Paz, A., Yehudayoff, A.: Distributed construction of purely additive spanners. In: 30th International Symposium on Distributed Computing, DISC, pp. 129–142 (2016)Google Scholar
  15. 15.
    Chechik, S.: Compact routing schemes with improved stretch. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 33–41 (2013)Google Scholar
  16. 16.
    Chechik, S.: New additive spanners. In: 24th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, pp. 498–512 (2013)Google Scholar
  17. 17.
    Coppersmith, D., Elkin, M.: Sparse sourcewise and pairwise distance preservers. SIAM J. Discrete Math. 20(2), 463–501 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cygan, M., Grandoni, F., Kavitha, T.: On pairwise spanners. In: 30th International Symposium on Theoretical Aspects of Computer Science, STACS, pp. 209–220 (2013)Google Scholar
  19. 19.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Derbel, B., Gavoille, C.: Fast deterministic distributed algorithms for sparse spanners. Theor. Comput. Sci. 399(1–2), 83–100 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Derbel, B., Gavoille, C., Peleg, D.: Deterministic distributed construction of linear stretch spanners in polylogarithmic time. In: 21st International Symposium on Distributed Computing, DISC, pp. 179–192 (2007)Google Scholar
  22. 22.
    Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: 27th Annual ACM Symposium on Principles of Distributed Computing, PODC, pp. 273–282 (2008)Google Scholar
  23. 23.
    Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: Local computation of nearly additive spanners. In: 23rd International Symposium on Distributed Computing, DISC, pp. 176–190 (2009)Google Scholar
  24. 24.
    Dor, D., Halperin, S., Zwick, U.: All-pairs almost shortest paths. SIAM J. Comput. 29(5), 1740–1759 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Drucker, A., Kuhn, F., Oshman, R.: On the power of the congested clique model. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 367–376 (2014)Google Scholar
  26. 26.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Elkin, M.: Computing almost shortest paths. ACM Trans. Algorithms 1(2), 283–323 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Elkin, M.: A near-optimal distributed fully dynamic algorithm for maintaining sparse spanners. In: 26th Annual ACM Symposium on Principles of Distributed Computing, PODC, pp. 185–194 (2007)Google Scholar
  29. 29.
    Elkin, M.: Streaming and fully dynamic centralized algorithms for constructing and maintaining sparse spanners. In: 34th International Colloquium on Automata, Languages and Programming, ICALP, pp. 716–727 (2007)Google Scholar
  30. 30.
    Elkin, M., Peleg, D.: (\(1+\epsilon, \beta \))-spanner constructions for general graphs. SIAM J. Comput. 33(3), 608–631 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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)CrossRefzbMATHGoogle Scholar
  32. 32.
    Erdős, P.: Extremal problems in graph theory. In: Theory of Graphs and Its Applications: Proceedings of the Symposium Held in Smolenice in June 1963, pp. 29–36. Pub. House of the Czechoslovak Academy of Sciences (1964)Google Scholar
  33. 33.
    Frischknecht, S., Holzer, S., Wattenhofer, R.: Networks cannot compute their diameter in sublinear time. In: 23rd Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, pp. 1150–1162 (2012)Google Scholar
  34. 34.
    Ghaffari, M., Kuhn, F.: Distributed minimum cut approximation. In: 27th International Symposium on Distributed Computing, DISC, pp. 1–15 (2013)Google Scholar
  35. 35.
    Holzer, S., Pinsker, N.: Approximation of distances and shortest paths in the broadcast congest clique. CoRR, abs/1412.3445 (2014)Google Scholar
  36. 36.
    Holzer, S., Wattenhofer, R.: Optimal distributed all pairs shortest paths and applications. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 355–364 (2012)Google Scholar
  37. 37.
    Kavitha, T.: New pairwise spanners. In: 32nd International Symposium on Theoretical Aspects of Computer Science, STACS, pp. 513–526 (2015)Google Scholar
  38. 38.
    Kavitha, T., Varma, N.M.: Small stretch pairwise spanners. In: 40th International Colloquium on Automata, Languages, and Programming, ICALP, pp. 601–612 (2013)Google Scholar
  39. 39.
    Knudsen, M.B.T.: Additive spanners: A simple construction. In: 14th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT, pp. 277–281 (2014)Google Scholar
  40. 40.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, New York (1997)CrossRefzbMATHGoogle Scholar
  41. 41.
    Lenzen, C., Peleg, D.: Efficient distributed source detection with limited bandwidth. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 375–382 (2013)Google Scholar
  42. 42.
    Matousek, J.: Lectures on Discrete Geometry. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  43. 43.
    Mitzenmacher, M., Upfal, E.: Probability and Computing-Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  44. 44.
    Parter, M.: Bypassing erdős’ girth conjecture: Hybrid stretch and sourcewise spanners. In: 41st International Colloquium on Automata, Languages, and Programming, ICALP, pp. 608–619 (2014)Google Scholar
  45. 45.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  46. 46.
    Peleg, D., Rubinovich, V.: A near-tight lower bound on the time complexity of distributed MST construction. In: 40th Annual Symposium on Foundations of Computer Science, FOCS, pp. 253–261 (1999)Google Scholar
  47. 47.
    Peleg, D., Schäffer, A.A.: Graph spanners. J. Graph Theory 13(1), 99–116 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Peleg, D., Ullman, J.D.: An optimal synchronizer for the hypercube. SIAM J. Comput. 18(4), 740–747 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Peleg, D., Upfal, E.: A trade-off between space and efficiency for routing tables. J. ACM 36(3), 510–530 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Peleg, D., Roditty, L., Tal, E.: Distributed algorithms for network diameter and girth. In: Automata, Languages, and Programming—39th International Colloquium, ICALP, pp. 660–672 (2012)Google Scholar
  51. 51.
    Pettie, S.: Low distortion spanners. ACM Trans. Algorithms 6(1), 7:1–7:22 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Pettie, S.: Distributed algorithms for ultrasparse spanners and linear size skeletons. Distrib. Comput. 22(3), 147–166 (2010)CrossRefzbMATHGoogle Scholar
  53. 53.
    Roditty, L., Zwick, U.: On dynamic shortest paths problems. Algorithmica 61(2), 389–401 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Roditty, L., Thorup, M., Zwick, U.: Deterministic constructions of approximate distance oracles and spanners. In: 32nd International Colloquium on Automata, Languages and Programming, ICALP, pp. 261–272 (2005)Google Scholar
  55. 55.
    Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA, pp. 1–10 (2001)Google Scholar
  56. 56.
    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 11–24 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Thorup, M., Zwick, U.: Spanners and emulators with sublinear distance errors. In: 17th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, pp. 802–809 (2006)Google Scholar
  58. 58.
    Woodruff, D.P.: Additive spanners in nearly quadratic time. In: 37th International Colloquium on Automata, Languages and Programming, ICALP, pp. 463–474 (2010)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Keren Censor-Hillel
    • 1
  • Telikepalli Kavitha
    • 2
  • Ami Paz
    • 1
  • Amir Yehudayoff
    • 3
  1. 1.Department of Computer ScienceTechnionHaifaIsrael
  2. 2.Tata Institute of Fundamental ResearchMumbaiIndia
  3. 3.Department of MathematicsTechnionHaifaIsrael

Personalised recommendations