A General Graph Model for Representing Exact Communication Volume in Parallel Sparse Matrix–Vector Multiplication

  • Aleksandar Trifunović
  • William Knottenbelt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4263)


In this paper, we present a new graph model of sparse matrix decomposition for parallel sparse matrix–vector multiplication. Our model differs from previous graph-based approaches in two main respects. Firstly, our model is based on edge colouring rather than vertex partitioning. Secondly, our model is able to correctly quantify and minimise the total communication volume of the parallel sparse matrix–vector multiplication while maintaining the computational load balance across the processors. We show that our graph edge colouring model is equivalent to the fine-grained hypergraph partitioning-based sparse matrix decomposition model. We conjecture that the existence of such a graph model should lead to faster serial and parallel sparse matrix decomposition heuristics and associated tools.


Bipartite Graph Graph Model Vector Multiplication Incidence Matrix Sparse Matrix 
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.


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  1. 1.
    Bisseling, R.H.: Personal communication (January 2006)Google Scholar
  2. 2.
    Bisseling, R.H., Meesen, W.: Communication balancing in parallel sparse matrix–vector multiplication. Electronic Transactions on Numerical Analysis: Special Volume on Combinatorial Scientific Computing 21, 47–65 (2005)MATHMathSciNetGoogle Scholar
  3. 3.
    Bollobás, B.: Combinatorics. Cambridge University Press, Cambridge (1986)MATHGoogle Scholar
  4. 4.
    Çatalyürek, U.V., Aykanat, C.: Hypergraph Partitioning-based Decomposition for Parallel Sparse–Matrix Vector Multiplication. IEEE Transactions on Parallel and Distributed Systems 10(7), 673–693 (1999)CrossRefGoogle Scholar
  5. 5.
    Çatalyürek, U.V., Aykanat, C.: A fine-grain hypergraph model for 2D decomposition of sparse matrices. In: Proc. 8th International Workshop on Solving Irregularly Structured Problems in Parallel, San Francisco, USA (April 2001)Google Scholar
  6. 6.
    Çatalyürek, U.V., Aykanat, C.: PaToH: Partitioning Tool for Hypergraphs, Version 3.0 (2001)Google Scholar
  7. 7.
    Devine, K.D., Boman, E.G., Heaphy, R.T., Bisseling, R.H., Çatalyürek, U.V.: Parallel hypergraph partitioning for scientific computing. In: Proc. 20th IEEE International Parallel and Distributed Processing Symposium (2006)Google Scholar
  8. 8.
    Devine, K.D., Boman, E.G., Heaphy, R.T., Hendrickson, B.A., Teresco, J.D., Faik, J., Flaherty, J.E., Gervasio, L.G.: New Challenges in Dynamic Load Balancing. Applied Numerical Mathematics 52(2–3), 133–152 (2005)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Fiduccia, C.M., Mattheyses, R.M.: A Linear Time Heuristic For Improving Network Partitions. In: Proc. 19th IEEE Design Automation Conference, pp. 175–181 (1982)Google Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Co., New York (1979)MATHGoogle Scholar
  11. 11.
    Hendrickson, B.A.: Graph Partitioning and Parallel Solvers: Has the Emperor No Clothes. In: Ferreira, A., Rolim, J.D.P., Teng, S.-H. (eds.) IRREGULAR 1998. LNCS, vol. 1457, pp. 218–225. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  12. 12.
    Hendrickson, B.A., Kolda, T.G.: Partitioning Rectangular and Structurally Nonsymmetric Sparse Matrices for Parallel Processing. SIAM Journal of Scientific Computing 21(6), 248–272 (2000)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Karypis, G., Kumar, V.: A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM Journal on Scientific Computing 20(1), 359–392 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Karypis, G., Schloegel, K., Kumar, V.: ParMeTiS: Parallel Graph Partitioning and Sparse Matrix Ordering Library, Version 3.0. University of Minnesota (2002)Google Scholar
  15. 15.
    Trifunović, A.: Parallel Algorithms for Hypergraph Partitioning. PhD thesis, Imperial College London (February 2006)Google Scholar
  16. 16.
    Trifunović, A., Knottenbelt, W.J.: A Parallel Algorithm for Multilevel k-way Hypergraph Partitioning. In: Proc. 3rd International Symposium on Parallel and Distributed Computing, University College Cork, Ireland, July 2004, pp. 114–121 (2004)Google Scholar
  17. 17.
    Trifunović, A., Knottenbelt, W.J.: Towards a Parallel Disk-Based Algorithm for Multilevel k-way Hypergraph Partitioning. In: Proc. 5th Workshop on Parallel and Distributed Scientific and Engineering Computing (April 2004)Google Scholar
  18. 18.
    Uçar, B., Aykanat, C.: Encapsulating Multiple Communication-Cost Metrics in Partitioning Sparse Rectangular Matrices for Parallel Matrix–Vector Multiples. SIAM Journal on Scientific Computing 25(6), 1837–1859 (2004)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Vastenhouw, B., Bisseling, R.H.: A Two-Dimensional Data Distribution Method for Parallel Sparse Matrix–Vector Multiplication. SIAM Review 47(1), 67–95 (2005)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Walshaw, C., Cross, M., Everett, M.G.: Parallel dynamic graph partitioning for adaptive unstructured meshes. J. Parallel Distrib. Comput. 47(2), 102–108 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Aleksandar Trifunović
    • 1
  • William Knottenbelt
    • 1
  1. 1.Department of ComputingImperial College LondonLondonUK

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