Deterministic Local Algorithms, Unique Identifiers, and Fractional Graph Colouring

  • Henning Hasemann
  • Juho Hirvonen
  • Joel Rybicki
  • Jukka Suomela
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7355)


We show that for any α > 1 there exists a deterministic distributed algorithm that finds a fractional graph colouring of length at most α(Δ + 1) in any graph in one synchronous communication round; here Δ is the maximum degree of the graph. The result is near-tight, as there are graphs in which the optimal solution has length Δ + 1.

The result is, of course, too good to be true. The usual definitions of scheduling problems (fractional graph colouring, fractional domatic partition, etc.) in a distributed setting leave a loophole that can be exploited in the design of distributed algorithms: the size of the local output is not bounded. Our algorithm produces an output that seems to be perfectly good by the usual standards but it is impractical, as the schedule of each node consists of a very large number of short periods of activity.

More generally, the algorithm shows that when we study distributed algorithms for scheduling problems, we can choose virtually any trade-off between the following three parameters: T, the running time of the algorithm, ℓ, the length of the schedule, and κ, the maximum number of periods of activity for a any single node. Here ℓ is the objective function of the optimisation problem, while κ captures the “subjective” quality of the solution. If we study, for example, bounded-degree graphs, we can trivially keep T and κ constant, at the cost of a large ℓ, or we can keep κ and ℓ constant, at the cost of a large T. Our algorithm shows that yet another trade-off is possible: we can keep T and ℓ constant at the cost of a large κ.


Schedule Problem Local Algorithm Local Output Graph Problem Vertex Colouring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bollobás, B.: The independence ratio of regular graphs. Proceedings of the American Mathematical Society 83(2), 433–436 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cole, R., Vishkin, U.: Deterministic coin tossing with applications to optimal parallel list ranking. Information and Control 70(1), 32–53 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Göös, M., Hirvonen, J., Suomela, J.: Lower bounds for local approximation. In: Proc. 31st Symposium on Principles of Distributed Computing, PODC 2012 (2012)Google Scholar
  4. 4.
    Hirvonen, J., Suomela, J.: Distributed maximal matching: greedy is optimal. In: Proc. 31st Symposium on Principles of Distributed Computing, PODC 2012 (2012)Google Scholar
  5. 5.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: What cannot be computed locally! In: Proc. 23rd Symposium on Principles of Distributed Computing (PODC 2004), pp. 300–309. ACM Press, New York (2004)Google Scholar
  6. 6.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: The price of being near-sighted. In: Proc. 17th Symposium on Discrete Algorithms (SODA 2006), pp. 980–989. ACM Press, New York (2006)CrossRefGoogle Scholar
  7. 7.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: Local computation: Lower and upper bounds (2010) (manuscript) arXiv:1011.5470 [cs.DC]Google Scholar
  8. 8.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM Journal on Computing 24(6), 1259–1277 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia (2000)zbMATHCrossRefGoogle Scholar
  10. 10.
    Shearer, J.B.: A note on the independence number of triangle-free graphs. Discrete Mathematics 46(1), 83–87 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Suomela, J.: Survey of local algorithms. ACM Computing Surveys (2011), (to appear)

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Henning Hasemann
    • 1
  • Juho Hirvonen
    • 2
  • Joel Rybicki
    • 2
  • Jukka Suomela
    • 2
  1. 1.Institute of Operating Systems and Computer NetworksTU BraunschweigGermany
  2. 2.Helsinki Institute for Information Technology HIIT, Department of Computer ScienceUniversity of HelsinkiFinland

Personalised recommendations