pp 1–15 | Cite as

A Simple Greedy Algorithm for Dynamic Graph Orientation

  • Edvin BerglinEmail author
  • Gerth Stølting Brodal
Part of the following topical collections:
  1. Special Issue: Algorithms and Computation


Graph orientations with low out-degree are one of several ways to efficiently store sparse graphs. If the graphs allow for insertion and deletion of edges, one may have to flip the orientation of some edges to prevent blowing up the maximum out-degree. We use arboricity as our sparsity measure. With an immensely simple greedy algorithm, we get parametrized trade-off bounds between out-degree and worst case number of flips, which previously only existed for amortized number of flips. We match the previous best worst-case algorithm (in \(\mathcal {O}\left( \log n\right) \) flips) for almost all values of arboricity and beat it for either constant or super-logarithmic arboricity. We also match a previous best amortized result for at least logarithmic arboricity, and give the first results with worst-case \(\mathcal {O}\left( 1\right) \) and \(\mathcal {O}\left( \sqrt{\log n}\right) \) flips nearly matching out-degree bounds to their respective amortized solutions.


Dynamic graph algorithms Graph arboricity Edge orientations 



  1. 1.
    Berglin, E.: Geometric covers, graph orientations, counter games. Ph.D. thesis, Aarhus University (2017)Google Scholar
  2. 2.
    Bernstein, A., Stein, C.: Fully dynamic matching in bipartite graphs. In: International Colloquium on Automata, Languages, and Programming, pp. 167–179. Springer (2015)Google Scholar
  3. 3.
    Bernstein, A., Stein, C.: Faster fully dynamic matchings with small approximation ratios. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 692–711. Society for Industrial and Applied Mathematics (2016)Google Scholar
  4. 4.
    Brodal, G.S., Fagerberg, R.: Dynamic representation of sparse graphs. In: Proceedings 6th International Workshop on Algorithms and Data Structures (WADS), Lecture Notes in Computer Science, vol. 1663, pp. 342–351. Springer (1999)Google Scholar
  5. 5.
    Dietz, P., Sleator, D.: Two algorithms for maintaining order in a list. In: Proceedings 19th Annual ACM Symposium on Theory of Computing (STOC), pp. 365–372. ACM (1987)Google Scholar
  6. 6.
    He, M., Tang, G., Zeh, N.: Orienting dynamic graphs, with applications to maximal matchings and adjacency queries. In: Proceedings 25th International Symposium on Algorithms and Computation (ISAAC), Lecture Notes in Computer Science, vol. 8889, pp. 128–140. Springer (2014)Google Scholar
  7. 7.
    Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM J. Discrete Math. 5(4), 596–603 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kopelowitz, T., Krauthgamer, R., Porat, E., Solomon, S.: Orienting fully dynamic graphs with worst-case time bounds. In: Proceedings 41st International Colloquium Automata, Languages, and Programming (ICALP), Part II, Lecture Notes in Computer Science, vol. 8573, pp. 532–543. Springer (2014)Google Scholar
  9. 9.
    Kowalik, Ł.: Adjacency queries in dynamic sparse graphs. Inf. Process. Lett. 102(5), 191–195 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Levcopoulos, C., Overmars, M.H.: A balanced search tree with \(O(1)\) worst-case update time. Acta Inform. 26(3), 269–277 (1988)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Neiman, O., Solomon, S.: Simple deterministic algorithms for fully dynamic maximal matching. ACM Trans. Algorithms 12(1), 7 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Peleg, D., Solomon, S.: Dynamic (\(1+ \varepsilon \))-approximate matchings: a density-sensitive approach. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 712–729. Society for Industrial and Applied Mathematics (2016)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aarhus UniversityAarhusDenmark

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