Distributed construction of purely additive spanners

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.

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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)

  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)

  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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

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

    MathSciNet  Article  MATH  Google 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)

  11. 11.

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

    MathSciNet  Article  MATH  Google 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)

  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)

  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)

  15. 15.

    Chechik, S.: Compact routing schemes with improved stretch. In: ACM Symposium on Principles of Distributed Computing, PODC, pp. 33–41 (2013)

  16. 16.

    Chechik, S.: New additive spanners. In: 24th Annual ACM–SIAM Symposium on Discrete Algorithms, SODA, pp. 498–512 (2013)

  17. 17.

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

    MathSciNet  Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

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

    MathSciNet  Article  MATH  Google 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)

  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)

  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)

  24. 24.

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

    MathSciNet  Article  MATH  Google 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)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

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

    MathSciNet  Article  MATH  Google 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)

  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)

  30. 30.

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

    MathSciNet  Article  MATH  Google 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)

    Article  MATH  Google 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)

  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)

  34. 34.

    Ghaffari, M., Kuhn, F.: Distributed minimum cut approximation. In: 27th International Symposium on Distributed Computing, DISC, pp. 1–15 (2013)

  35. 35.

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

  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)

  37. 37.

    Kavitha, T.: New pairwise spanners. In: 32nd International Symposium on Theoretical Aspects of Computer Science, STACS, pp. 513–526 (2015)

  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)

  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)

  40. 40.

    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, New York (1997)

    Google 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)

  42. 42.

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

    Google Scholar 

  43. 43.

    Mitzenmacher, M., Upfal, E.: Probability and Computing-Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)

    Google 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)

  45. 45.

    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics, Philadelphia (2000)

    Google 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)

  47. 47.

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

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

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

    MathSciNet  Article  MATH  Google Scholar 

  49. 49.

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

    MathSciNet  Article  MATH  Google 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)

  51. 51.

    Pettie, S.: Low distortion spanners. ACM Trans. Algorithms 6(1), 7:1–7:22 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  52. 52.

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

    Article  MATH  Google Scholar 

  53. 53.

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

    MathSciNet  Article  MATH  Google 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)

  55. 55.

    Thorup, M., Zwick, U.: Compact routing schemes. In: SPAA, pp. 1–10 (2001)

  56. 56.

    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 11–24 (2005)

    MathSciNet  Article  MATH  Google 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)

  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)

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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.

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Correspondence to Ami Paz.

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

Amir Yehudayoff: Research supported by ISF Grant 1162/15.

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Censor-Hillel, K., Kavitha, T., Paz, A. et al. Distributed construction of purely additive spanners. Distrib. Comput. 31, 223–240 (2018). https://doi.org/10.1007/s00446-017-0306-2

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