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Derandomizing local distributed algorithms under bandwidth restrictions

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Abstract

This paper addresses the cornerstone family of local problems in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the n nodes can only send \(O(\log n)\)-bit messages in each round of communication. Our framework mostly follows by the derandomization approach of Luby (J Comput Syst Sci 47(2):250–286, 1993) combined with the power of all to all communication. Our key results are as follows: first, we show that in the congested clique model, which allows all-to-all communication, there is a deterministic maximal independent set algorithm that runs in \(O(\log ^2 {\varDelta })\) rounds, where \({\varDelta }\) is the maximum degree. When \({\varDelta }=O(n^{1/3})\), the bound improves to \(O(\log {\varDelta })\). In addition, we deterministically construct a \((2k-1)\)-spanner with \(O(kn^{1+1/k}\log n)\) edges in \(O(k \log n)\) rounds in the congested clique model.

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Notes

  1. As standard, with high probability means with probability that is at least \(1-1/n^c\) for a constant c.

  2. Flipping a biased coin with probability \(1/2^i\), is the same as getting a uniformly distributed number y in [1, b] and outputting 1 if and only if \(y \in [1, 2^{b-i}]\).

  3. The term \(M_{t,b}(u,W(v))\) is important as it is what prevents double counting, because the corresponding random variables defined by the neighbors of v are mutually exclusive.

  4. The simplified version that we describe consists of only one phase that includes both phases mentioned in [8]. This is because the number of clusters in the last iteration k is forced to be zero and hence all vertices are unclustered in the last iteration and so they connect to all clusters of the previous iteration.

References

  1. Alon, N., Babai, L., Itai, A.: A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algorithms 7(4), 567–583 (1986)

    Article  MathSciNet  Google Scholar 

  2. Awerbuch, B., Goldberg, A.V., Luby, M., Plotkin, S.A.: Network decomposition and locality in distributed computation. In: FOCS, pp. 364–369 (1989)

  3. Barenboim, L.: Deterministic (\(\Delta + 1\))-coloring in sublinear (in \(\Delta \)) time in static, dynamic and faulty networks. In: PODC, pp. 345–354 (2015)

  4. Barenboim, L., Elkin, M.: Distributed graph coloring: fundamentals and recent developments. Synth. Lect. Distrib. Comput. Theory 4(1), 1–171 (2013)

    Article  Google Scholar 

  5. Barenboim, L., Elkin, M., Gavoille, C.: A fast network-decomposition algorithm and its applications to constant-time distributed computation. In: SIROCCO, pp. 209–223 (2015)

  6. Barenboim, L., Elkin, M., Kuhn, F.: Distributed (\(\Delta + 1\))-coloring in linear (in \(\Delta \)) time. SIAM J. Comput. 43(1), 72–95 (2014)

    Article  MathSciNet  Google Scholar 

  7. Barenboim, L., Elkin, M., Pettie, S., Schneider, J.: The locality of distributed symmetry breaking. In: FOCS, pp. 321–330 (2012)

  8. 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)

    Article  MathSciNet  Google Scholar 

  9. Benjamini, I., Gurel-Gurevich, O., Peled, R.: On k-wise independent distributions and Boolean functions. arXiv preprint arXiv:1201.3261 (2012)

  10. Berger, B., Rompel, J.: Simulating (log CN)-wise independence in NC. J. ACM (JACM) 38(4), 1026–1046 (1991)

    Article  Google Scholar 

  11. Brandt, S., Fischer, O., Hirvonen, J., Keller, B., Lempiäinen, T., Rybicki, J., Suomela, J., Uitto, J.: A lower bound for the distributed Lovász local lemma. In: STOC, pp. 479–488 (2016)

  12. Censor-Hillel, K., Kaski, P., Korhonen, J.H., Lenzen, C., Paz, A., Suomela, J.: Algebraic methods in the congested clique. In: PODC, pp. 143–152 (2015)

  13. Chandrasekaran, K., Goyal, N., Haeupler, B.: Deterministic algorithms for the Lovász local lemma. SIAM J. Comput. 42(6), 2132–2155 (2013)

    Article  MathSciNet  Google Scholar 

  14. Chang, Y.-J., Fischer, M., Ghaffari, M., Uitto, J., Zheng, Y.: The complexity of (\(\Delta +1\)) coloring in congested clique, massively parallel computation, and centralized local computation. In: Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC 2019, Toronto, ON, Canada, July 29–August 2, 2019, pp. 471–480 (2019)

  15. Chang, Y.-J., Kopelowitz, T., Pettie, S.: An exponential separation between randomized and deterministic complexity in the LOCAL model. In: FOCS, pp. 615–624 (2016)

  16. Derbel, B., Gavoille, C.: Fast deterministic distributed algorithms for sparse spanners. In: SIROCCO, pp. 100–114 (2006)

  17. Derbel, B. Gavoille, C., Peleg, D.: Deterministic distributed construction of linear stretch spanners in polylogarithmic time. In: DISC, pp. 179–192 (2007)

  18. Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: On the locality of distributed sparse spanner construction. In: PODC, pp. 273–282 (2008)

  19. Derbel, B., Gavoille, C., Peleg, D., Viennot, L.: Local computation of nearly additive spanners. In: DISC, pp. 176–190 (2009)

  20. Derbel, B., Mosbah, M., Zemmari, A.: Sublinear fully distributed partition with applications. Theory Comput. Syst. 47(2), 368–404 (2010)

    Article  MathSciNet  Google Scholar 

  21. Dolev, D., Lenzen, C., Peled, S.: “tri, tri again”: finding triangles and small subgraphs in a distributed setting—(extended abstract). In: DISC, pp. 195–209 (2012)

  22. Drucker, A., Kuhn, F., Oshman, R.: On the power of the congested clique model. In: PODC, pp. 367–376 (2014)

  23. Elkin, M.: A near-optimal distributed fully dynamic algorithm for maintaining sparse spanners. In: PODC, pp. 185–194 (2007)

  24. Feuilloley, L., Fraigniaud, P.: Randomized local network computing. In: SPAA, pp. 340–349 (2015)

  25. Le Gall, F.: Further algebraic algorithms in the congested clique model and applications to graph-theoretic problems. In: DISC, pp. 57–70 (2016)

  26. Ghaffari, M.: An improved distributed algorithm for maximal independent set. In: SODA, pp. 270–277 (2016)

  27. Ghaffari, M.: Distributed MIS via all-to-all communication. In: Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC 2017, Washington, DC, USA, July 25–27, 2017, pp. 141–149 (2017)

  28. Ghaffari, M., Parter, M.: MST in log-star rounds of congested clique. In: PODC, pp. 19–28 (2016)

  29. Goldberg, M., Spencer, T.: A new parallel algorithm for the maximal independent set problem. SIAM J. Comput. 18(2), 419–427 (1989)

    Article  MathSciNet  Google Scholar 

  30. Han, Y.: A fast derandomization scheme and its applications. SIAM J. Comput. 25(1), 52–82 (1996)

    Article  MathSciNet  Google Scholar 

  31. Hegeman, J.W., Pandurangan, G., Pemmaraju, S.V., Sardeshmukh, V.B., Scquizzato, M.: Toward optimal bounds in the congested clique: graph connectivity and MST. In: PODC, pp. 91–100 (2015)

  32. Hegeman, J.W., Pemmaraju, S.V.: Lessons from the congested clique applied to mapreduce. In: SIROCCO, pp. 149–164 (2014)

  33. Hegeman, J.W., Pemmaraju, S.V., Sardeshmukh, V.: Near-constant-time distributed algorithms on a congested clique. In: DISC, pp. 514–530 (2014)

  34. Henzinger, M., Krinninger, S., Nanongkai, D.: A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. In: STOC, pp. 489–498 (2016)

  35. Holzer, S., Pinsker, N.: Approximation of distances and shortest paths in the broadcast congest clique. In: OPODIS, pp. 6:1–6:16 (2015)

  36. Israeli, A., Itai, A.: A fast and simple randomized parallel algorithm for maximal matching. Inf. Process. Lett. 22(2), 77–80 (1986)

    Article  MathSciNet  Google Scholar 

  37. Jurdzinski, T., Nowicki, K.: MST in O(1) rounds of congested clique. In: SODA, pp. 2620–2632. SIAM (2018)

  38. Karp, R.M., Wigderson, A.: A fast parallel algorithm for the maximal independent set problem. In: STOC, pp. 266–272 (1984)

  39. Kuhn, F., Moscibroda, T., Wattenhofer, R.: Local computation: lower and upper bounds. J. ACM 63(2), 17 (2016)

    Article  MathSciNet  Google Scholar 

  40. Lenzen, C.: Optimal deterministic routing and sorting on the congested clique. In: PODC, pp. 42–50 (2013)

  41. Lenzen, C., Wattenhofer, R.: Tight bounds for parallel randomized load balancing: extended abstract. In: STOC, pp. 11–20 (2011)

  42. Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)

    Article  MathSciNet  Google Scholar 

  43. Lotker, Z., Pavlov, E., Patt-Shamir, B., Peleg, D.: MST construction in O(\(\log log n\)) communication rounds. In: SPAA, pp. 94–100 (2003)

  44. Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15(4), 1036–1053 (1986)

    Article  MathSciNet  Google Scholar 

  45. Luby, M.: Removing randomness in parallel computation without a processor penalty. J. Comput. Syst. Sci. 47(2), 250–286 (1993)

    Article  MathSciNet  Google Scholar 

  46. Motwani, R., Naor, J., Naor, M.: The probabilistic method yields deterministic parallel algorithms. J. Comput. Syst. Sci. 49(3), 478–516 (1994)

    Article  MathSciNet  Google Scholar 

  47. Nanongkai, D.: Distributed approximation algorithms for weighted shortest paths. In: STOC, pp. 565–573 (2014)

  48. Naor, M., Stockmeyer, L.J.: What can be computed locally? SIAM J. Comput. 24(6), 1259–1277 (1995)

    Article  MathSciNet  Google Scholar 

  49. Panconesi, A., Srinivasan, A.: Improved distributed algorithms for coloring and network decomposition problems. In: STOC, pp. 581–592 (1992)

  50. Pantziou, G., Spirakis, P., Zaroliagis, C.: Fast parallel approximations of the maximum weighted cut problem through derandomization. In: FSTTCS, pp. 20–29 (1989)

  51. Parter, M.: (\(\Delta + 1\)) coloring in the congested clique model. In: 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9–13, 2018, Prague, Czech Republic, pp. 160:1–160:14 (2018)

  52. Parter, M., Su, H.-H.: Randomized (\(\Delta + 1\))-coloring in \(O(\log *\Delta )\) congested clique rounds. In: 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15–19, 2018, pp. 39:1–39:18 (2018)

  53. Patt-Shamir, B., Teplitsky, M.: The round complexity of distributed sorting: extended abstract. In: PODC, pp. 249–256 (2011)

  54. Pemmaraju, S.V.: Equitable coloring extends Chernoff-hoeffding bounds. In: Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques, 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2001 and 5th International Workshop on Randomization and Approximation Techniques in Computer Science, RANDOM 2001 Berkeley, CA, USA, August 18–20, 2001, Proceedings, pp. 285–296 (2001). https://doi.org/10.1007/3-540-44666-4_31

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

    Article  Google Scholar 

  56. Roditty, L., Thorup, M., Zwick, U. Deterministic constructions of approximate distance oracles and spanners. In: ICALP, pp. 261–272 (2005)

  57. Schmidt, J.P., Siegel, A., Srinivasan, A.: Chernoff-hoeffding bounds for applications with limited independence. SIAM J. Discrete Math. 8(2), 223–250 (1995)

    Article  MathSciNet  Google Scholar 

  58. Srivastav, A., Kliemann, L.: Parallel algorithms via the probabilistic method. In: Rajasekaran, S. (ed.) Handbook of Parallel Computing: Models. Algorithms and Applications. Chapman and Hall/CRC, Boca Raton (2007)

    Google Scholar 

  59. Suomela, J.: Survey of local algorithms. ACM Comput. Surv. 45(2), 24 (2013)

    Article  Google Scholar 

  60. Vadhan, S.P.: Pseudorandomness. Found. Trends Theor. Comput. Sci. 7(1–3), 1–336 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We are very grateful to Mohsen Ghaffari for many helpful discussions and useful observations involving the derandomization of his MIS algorithm.

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Authors and Affiliations

  1. Technion, Haifa, Israel

    Keren Censor-Hillel

  2. Weizmann Institute, Rehovot, Israel

    Merav Parter

  3. NII, Tokyo, Japan

    Gregory Schwartzman

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  1. Keren Censor-Hillel
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  2. Merav Parter
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  3. Gregory Schwartzman
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Correspondence to Merav Parter.

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Keren Censor-Hillel: Supported in part by the Israel Science Foundation (Grant 1696/14). This project has received funding from the European Union’s Horizon 2020 Research And Innovation Program under grant Agreement No. 755839. Gregory Schwartzman: This work was supported by JSPS KAKENHI Grant Numbers 19K20216 and JP18H05291.

Pseudocode of the deterministic MIS algorithm

Pseudocode of the deterministic MIS algorithm

Let \({\mathcal {H}}={\mathcal {H}}_{\gamma ,\beta }\) with \(\gamma ={\varTheta }(\log n)\) and \(\beta ={\varTheta }(\log {\varDelta })\) be given by Lemma 1. Let \({\mathcal {H}}(Y_i) \subseteq {\mathcal {H}}_{\gamma ,\beta }\) be the collection of all hash functions that agree with the partial seed \(Y_i\). Each function \(h \in {\mathcal {H}}(Y_i)\) corresponds to a deterministic MIS algorithm.

For a hash function \(h \in {\mathcal {H}}\) and value \(p=1/2^i\) representing the probability of a node to be marked, define \(m_h(v,p)=1\) if \(h(ID(v)) \in [1,2^{\beta -i}]\) and \(m_h(v,p)=0\) otherwise. That is, marking a node v with probability p is simulated deterministically by computing \(m_h(v,p)\), since we output 1 only if the value h(ID(v)) appears in the top \(1/2^i\) fraction of the range \([1, \beta ]\). For \(p=1/2^i\) and \(p'=1/2^{i'}\), define \(m_h(v,u, p,p')=1\) if \(h(ID(v)) \in [1,2^{\beta -i}]\) and \(h(ID(u)) \in [1, 2^{\beta -i'}]\) and \(m_h(v,u, p,p')=0\) otherwise. That is \(m_h(v,u, p,p')=1\) is the deterministic simulation of having both v and u being marked when v, u are marked with probability \(p,p'\). Let \(\alpha \in (0,1]\) be the constant such that every golden node is removed with probability at least \(\alpha \) when using pairwise independence. Algorithm 5 gives the pseudocode of our deterministic MIS algorithm.

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Censor-Hillel, K., Parter, M. & Schwartzman, G. Derandomizing local distributed algorithms under bandwidth restrictions. Distrib. Comput. 33, 349–366 (2020). https://doi.org/10.1007/s00446-020-00376-1

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