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Improved distributed degree splitting and edge coloring

  • Mohsen Ghaffari
  • Juho Hirvonen
  • Fabian KuhnEmail author
  • Yannic Maus
  • Jukka Suomela
  • Jara Uitto
Article
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Abstract

The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires orienting the edges such that each node has almost the same number of incoming and outgoing edges, again up to a small additive discrepancy. We present deterministic distributed algorithms for both variants, which improve on their counterparts presented by Ghaffari and Su (Proc SODA 2017:2505–2523, 2017): our algorithms are significantly simpler and faster, and have a much smaller discrepancy. This also leads to a faster and simpler deterministic algorithm for \((2+o(1))\varDelta \)-edge-coloring, improving on that of Ghaffari and Su.

Keywords

Distributed graph algorithms Degree splitting Edge coloring Discrepancy 

Notes

References

  1. 1.
    Barenboim, L., Elkin, M.: Distributed deterministic edge coloring using bounded neighborhood independence. Proc. PODC 2011, 129–138 (2011)zbMATHGoogle Scholar
  2. 2.
    Beck, J., Fiala, T.: “Integer-making” theorems. Discrete Appl. Math. 3(1), 1–8 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bednarchak, D., Helm, M.: A note on the Beck–Fiala theorem. Combinatorica 17(1), 147–149 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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. Proc. STOC 2016, 479–488 (2016)zbMATHGoogle Scholar
  5. 5.
    Bukh, B.: An improvement of the Beck–Fiala theorem. Comb. Probab. Comput. 25(03), 380–398 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chang, Y.J., Kopelowitz, T., Pettie, S.: An exponential separation between randomized and deterministic complexity in the local model. Proc. FOCS 2016, 615–624 (2016)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chazelle, B.: The Discrepancy Method: Randomness and Complexity. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  8. 8.
    Czygrinow, A., Hańćkowiak, M., Karoński, M.: Distributed \(O(\Delta \log n)\)-edge-coloring algorithm. Proc. ESA 2001, 345–355 (2001)zbMATHGoogle Scholar
  9. 9.
    Dinitz, Y.: Dinitz’ algorithm: the original version and Even’s version. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science, Essays in Memory of Shimon Even, pp. 218–240. Springer, Berlin (2006)CrossRefGoogle Scholar
  10. 10.
    Fischer, M., Ghaffari, M., Kuhn, F.: Deterministic distributed edge-coloring via hypergraph maximal matching. In: 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15–17, 2017, pp. 180–191 (2017)Google Scholar
  11. 11.
    Ghaffari, M., Harris, D.G., Kuhn, F.: On derandomizing local distributed algorithms. In: 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS (2018)Google Scholar
  12. 12.
    Ghaffari, M., Hirvonen, J., Kuhn, F., Maus, Y., Suomela, J., Uitto, J.: Improved distributed degree splitting and edge coloring. In: Proceedings of the 31st Symposium on Distributed Computing (DISC), pp. 19:1–19:15 (2017)Google Scholar
  13. 13.
    Ghaffari, M., Kuhn, F., Maus, Y., Uitto, J.: Deterministic distributed edge-coloring with fewer colors. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25–29, 2018, pp. 418–430 (2018)Google Scholar
  14. 14.
    Ghaffari, M., Su, H.H.: Distributed degree splitting, edge coloring, and orientations. Proc. SODA 2017, 2505–2523 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hańćkowiak, M., Karoński, M., Panconesi, A.: On the distributed complexity of computing maximal matchings. SIAM J. Discrete Math. 15(1), 41–57 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Harris, D.G.: Distributed approximation algorithms for maximum matching in graphs and hypergraphs. ArXiv e-prints (2018)Google Scholar
  17. 17.
    Israeli, A., Shiloach, Y.: An improved parallel algorithm for maximal matching. Inf. Process. Lett. 22(2), 57–60 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Karloff, H.J., Shmoys, D.B.: Efficient parallel algorithms for edge coloring problems. J. Algorithms 8(1), 39–52 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Linial, N.: Distributive graph algorithms—global solutions from local data. Proc. FOCS 1987, 331–335 (1987)Google Scholar
  20. 20.
    Naor, M., Stockmeyer, L.: What can be computed locally? Proc. STOC 1993, 184–193 (1993)zbMATHGoogle Scholar
  21. 21.
    Panconesi, A., Rizzi, R.: Some simple distributed algorithms for sparse networks. Distrib. Comput. 14(2), 97–100 (2001)CrossRefGoogle Scholar
  22. 22.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.ETH ZurichZurichSwitzerland
  2. 2.Aalto UniversityHelsinkiFinland
  3. 3.University of FreiburgFreiburgGermany

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