Tree-Based Coarsening and Partitioning of Complex Networks

  • Roland Glantz
  • Henning Meyerhenke
  • Christian Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8504)


Many applications produce massive complex networks whose analysis would benefit from parallel processing. Parallel algorithms, in turn, often require a suitable network partition. For solving optimization tasks such as graph partitioning on large networks, multilevel methods are preferred in practice. Yet, complex networks pose challenges to established multilevel algorithms, in particular to their coarsening phase.

One way to specify a (recursive) coarsening of a graph is to rate its edges and then contract the edges as prioritized by the rating. In this paper we (i) define weights for the edges of a network that express the edges’ importance for connectivity, (ii) compute a minimum weight spanning tree T m w.r.t. these weights, and (iii) rate the network edges based on the conductance values of T m ’s fundamental cuts. To this end, we also (iv) develop the first optimal linear-time algorithm to compute the conductance values of all fundamental cuts of a given spanning tree.

We integrate the new edge rating into a leading multilevel graph partitioner and equip the latter with a new greedy postprocessing for optimizing the maximum communication volume (MCV). Bipartitioning experiments on established benchmark graphs show that both the postprocessing and the new edge rating improve upon the state of the art by more than 10%. In total, with a modest increase in running time, our new approach reduces the MCV of complex network partitions by 20.4%.


Graph coarsening multilevel graph partitioning complex networks fundamental cuts spanning trees 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bader, D.A., Meyerhenke, H., Sanders, P., Wagner, D.: Graph Partitioning and Graph Clustering – 10th DIMACS Impl. Challenge. Contemporary Mathematics, vol. 588. AMS (2013)Google Scholar
  2. 2.
    Bender, M.A., Farach-Colton, M.: The LCA problem revisited. In: Gonnet, G.H., Viola, A. (eds.) LATIN 2000. LNCS, vol. 1776, pp. 88–94. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Bichot, C., Siarry, P. (eds.): Graph Partitioning. Wiley (2011)Google Scholar
  4. 4.
    Buluç, A., Meyerhenke, H., Safro, I., Sanders, P., Schulz, C.: Recent Advances in Graph Partitioning. Technical Report ArXiv:1311.3144 (2014)Google Scholar
  5. 5.
    Chen, J., Safro, I.: Algebraic distance on graphs. SIAM J. Comput. 6, 3468–3490 (2011)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chevalier, C., Safro, I.: Comparison of coarsening schemes for multi-level graph partitioning. In: Proc. Learning and Intelligent Optimization (2009)Google Scholar
  7. 7.
    de Costa, L.F., Oliveira Jr., O.N., Travieso, G., Rodrigues, F.A., Boas, P.R.V., Antiqueira, L., Viana, M.P., Correa Rocha, L.E.: Analyzing and modeling real-world phenomena with complex networks: a survey of applications. Advances in Physics 60(3), 329–412 (2011)CrossRefGoogle Scholar
  8. 8.
    Fagginger Auer, B.O., Bisseling, R.H.: Graph coarsening and clustering on the GPU. In: Graph Partitioning and Graph Clustering. AMS and DIMACS (2013)Google Scholar
  9. 9.
    Felzenszwalb, P.F., Huttenlocher, D.P.: Efficient graph-based image segmentation. Int. J. Comput. Vision 59(2), 167–181 (2004)CrossRefGoogle Scholar
  10. 10.
    Fischer, J., Heun, V.: Theoretical and Practical Improvements on the RMQ-Problem, with Applications to LCA and LCE. In: Lewenstein, M., Valiente, G. (eds.) CPM 2006. LNCS, vol. 4009, pp. 36–48. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Glantz, R., Meyerhenke, H., Schulz, C.: Tree-based Coarsening and Partitioning of Complex Networks. Technical Report arXiv:1402.2782 (2014)Google Scholar
  12. 12.
    Grady, L., Schwartz, E.L.: Isoperimetric graph partitioning for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(3), 469–475 (2006)CrossRefGoogle Scholar
  13. 13.
    Hendrickson, B., Kolda, T.G.: Graph partitioning models for parallel computing. Parallel Computing 26(12), 1519–1534 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Holtgrewe, M., Sanders, P., Schulz, C.: Engineering a scalable high quality graph partitioner. In: 24th Int. Parallel and Distributed Processing Symp, IPDPS (2010)Google Scholar
  15. 15.
    Jungnickel, D.: Graphs, Networks and Algorithms, 2nd edn. Algorithms and Computation in Mathematics, vol. 5. Springer, Berlin (2005)zbMATHGoogle Scholar
  16. 16.
    Kannan, R., Vempala, S., Vetta, A.: On clusterings: Good, bad and spectral. J. of the ACM 51(3), 497–515 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Karypis, G., Kumar, V.: A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM J. on Scientific Computing 20(1), 359–392 (1998)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Leskovec, J.: Stanford Network Analysis Package (SNAP)Google Scholar
  19. 19.
    Meyerhenke, H., Monien, B., Schamberger, S.: Graph partitioning and disturbed diffusion. Parallel Computing 35(10-11), 544–569 (2009)CrossRefGoogle Scholar
  20. 20.
    Pritchard, D., Thurimella, R.: Fast computation of small cuts via cycle space sampling. ACM Trans. Algorithms 46, 46:1–46:30 (2011)Google Scholar
  21. 21.
    Safro, I., Sanders, P., Schulz, C.: Advanced coarsening schemes for graph partitioning. In: Klasing, R. (ed.) SEA 2012. LNCS, vol. 7276, pp. 369–380. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  22. 22.
    Sanders, P., Schulz, C.: KaHIP – Karlsruhe High Qualtity Partitioning Homepage,
  23. 23.
    Sanders, P., Schulz, C.: Think Locally, Act Globally: Highly Balanced Graph Partitioning. In: Bonifaci, V., Demetrescu, C., Marchetti-Spaccamela, A. (eds.) SEA 2013. LNCS, vol. 7933, pp. 164–175. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Schulz, C.: Hiqh Quality Graph Partititioning. PhD thesis, Karlsruhe Institute of Technology (2013)Google Scholar
  25. 25.
    Soper, A.J., Walshaw, C., Cross, M.: A combined evolutionary search and multilevel optimisation approach to graph partitioning. Journal of Global Optimization 29(2), 225–241 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Wassenberg, J., Middelmann, W., Sanders, P.: An efficient parallel algorithm for graph-based image segmentation. In: Jiang, X., Petkov, N. (eds.) CAIP 2009. LNCS, vol. 5702, pp. 1003–1010. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Roland Glantz
    • 1
  • Henning Meyerhenke
    • 1
  • Christian Schulz
    • 1
  1. 1.Karlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations