Advertisement

Computing the Degeneracy of Large Graphs

  • Martín Farach-Colton
  • Meng-Tsung Tsai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

Any ordering of the nodes of an n-node, m-edge simple undirected graph G defines an acyclic orientation of the edges in which each edge is oriented from the earlier node in the ordering to the later. The degeneracy on an ordering is the maximum outdegree it induces, and the degeneracy of a graph is smallest degeneracy of any node ordering. Small-degeneracy orderings have many applications.

We give an algorithm for generating an ordering whose degeneracy approximates the minimum possible, that is, it approximates the degeneracy of the graph. Although the optimal ordering itself can be computed in \(\mathcal{O}(m)\) time and \(\mathcal{O}(m)\) space, such algorithms are infeasible for large graphs. Our approximation algorithm is semi-streaming: it uses less space, can achieve a constant approximation ratio, and accesses the graph in logarithmic read-only passes.

Keywords

Greedy Algorithm Approximation Factor Large Graph Candidate Node Sequential Pass 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahn, K.J., Guha, S.: Linear programming in the semi-streaming model with application to the maximum matching problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 526–538. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Finding and counting given length cycles. Algorithmica 17(3), 209–223 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bar-Yossef, Z., Kumar, R., Sivakumar, D.: Reductions in streaming algorithms, with an application to counting triangles in graphs. In: Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2002, pp. 623–632. Society for Industrial and Applied Mathematics (2002)Google Scholar
  4. 4.
    Bollobás, B.: Extremal graph theory. Academic Press (1978)Google Scholar
  5. 5.
    Bollobás, B.: The evolution of sparse graphs. In: Graph Theory and Combinatorics, Proc. Cambridge Combinatorial Conf., pp. 35–57. Academic Press (1984)Google Scholar
  6. 6.
    Chakrabarti, A., Cormode, G., McGregor, A.: Annotations in data streams. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 222–234. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Charikar, M.: Greedy approximation algorithms for finding dense components in a graph. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 84–95. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Charikar, M., Chen, K., Farach-Colton, M.: Finding frequent items in data streams. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 693–703. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Cormode, G., Hadjieleftheriou, M.: Finding frequent items in data streams. Proc. VLDB Endow. 1(2), 1530–1541 (2008)Google Scholar
  10. 10.
    Cormode, G., Muthukrishnan, S.: An improved data stream summary: the count-min sketch and its applications. J. Algorithms 55(1), 58–75 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Dean, A.M., Hutchinson, J.P., Scheinerman, E.R.: On the thickness and arboricity of a graph. Journal of Combinatorial Theory, Series B 52(1), 147–151 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Dvořák, Z.: Constant-factor approximation of the domination number in sparse graphs. European Journal of Combinatorics 34(5), 833–840 (2013)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Feigenbaum, J., Kannan, S., McGregor, A., Suri, S., Zhang, J.: On graph problems in a semi-streaming model. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 531–543. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Frank, A., Gyarfas, A.: How to orient the edges of a graph. In: Combinatorics Volume I, Proc. of the Fifth Hungarian Colloquium on Combinatorics, vol. I, pp. 353–364 (1976)Google Scholar
  15. 15.
    Gabow, H., Westermann, H.: Forests, frames, and games: algorithms for matroid sums and applications. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC 1988 pp. 407–421. ACM (1988)Google Scholar
  16. 16.
    Goldberg, A.V.: Finding a maximum density subgraph. Tech. rep. (1984)Google Scholar
  17. 17.
    Kawano, S., Yamazaki, K.: Worst case analysis of a greedy algorithm for graph thickness. Information Processing Letters 85(6), 333–337 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kowalik, Ł.: Approximation scheme for lowest outdegree orientation and graph density measures. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 557–566. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Lenzen, C., Wattenhofer, R.: Minimum dominating set approximation in graphs of bounded arboricity. In: Lynch, N.A., Shvartsman, A.A. (eds.) DISC 2010. LNCS, vol. 6343, pp. 510–524. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Liu, P., Wang, D.W., Wu, J.J.: Efficient parallel i/o scheduling in the presence of data duplication. In: International Conference on Parallel Processing, pp. 231–238 (2003)Google Scholar
  21. 21.
    Mansfield, A.: Determining the thickness of graphs is NP-hard. Math. Proc. Cambridge Philos. Soc. 93, 9–23 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Matula, D.W., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. ACM 30(3), 417–427 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Muthukrishnan, S.: Data streams: Algorithms and applications. Tech. rep. (2003)Google Scholar
  24. 24.
    O’Connell, T.C.: A survey of graph algorithms under extended streaming models of computation. In: Fundamental Problems in Computing, pp. 455–476. Springer, Netherlands (2009)Google Scholar
  25. 25.
    Ruhl, J.M.: Efficient Algorithms for New Computational Models. Ph.D. thesis, Massachusetts Institute of Technology (September 2003)Google Scholar
  26. 26.
    Schank, T., Wagner, D.: Finding, counting and listing all triangles in large graphs, an experimental study. In: Nikoletseas, S.E. (ed.) WEA 2005. LNCS, vol. 3503, pp. 606–609. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  27. 27.
    Zhang, J.: A survey on streaming algorithms for massive graphs. In: Managing and Mining Graph Data, Advances in Database Systems, vol. 40, pp. 393–420. Springer, US (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Martín Farach-Colton
    • 1
  • Meng-Tsung Tsai
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

Personalised recommendations