Evaluating Non-square Sparse Bilinear Forms on Multiple Vector Pairs in the I/O-Model

  • Gero Greiner
  • Riko Jacob
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6281)


We consider evaluating one bilinear form defined by a sparse N y ×N x matrix A having h entries on w pairs of vectors The model of computation is the semiring I/O-model with main memory size M and block size B. For a range of low densities (small h), we determine the I/O-complexity of this task for all meaningful choices of N x , N y , w, M and B, as long as M ≥ B 2 (tall cache assumption). To this end, we present asymptotically optimal algorithms and matching lower bounds. Moreover, we show that multiplying the matrix A with w vectors has the same worst-case I/O-complexity.


Bilinear Form Sparse Matrix Internal Memory Matrix Vector Multiplication Matrix Vector 
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  1. 1.
    Aggarwal, A., Vitter, J.S.: The input/output complexity of sorting and related problems. Communications of the ACM 31(9), 1116–1127 (1988)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bender, M.A., Brodal, G.S., Fagerberg, R., Jacob, R., Vicari, E.: Optimal sparse matrix dense vector multiplication in the I/O-model. In: Proceedings of SPAA 2007, pp. 61–70. ACM, New York (2007)CrossRefGoogle Scholar
  3. 3.
    Greiner, G., Jacob, R.: Evaluating non-square sparse bilinear forms on multiple vector pairs in the I/O-model. Technical report, Technische Universität München (June 2010)Google Scholar
  4. 4.
    Greiner, G., Jacob, R.: The I/O complexity of sparse matrix dense matrix multiplication. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 143–156. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  5. 5.
    Hong, J.-W., Kung, H.T.: I/O complexity: The red-blue pebble game. In: Proceedings of STOC 1981, pp. 326–333. ACM, New York (1981)Google Scholar
  6. 6.
    Jacob, R., Schnupp, M.: Experimental performance of I/O-optimal sparse matrix dense vector multiplication algorithms within main memory. Technical report, Technische Universität München (June 2010)Google Scholar
  7. 7.
    Lieber, T.: Combinatorial approaches to optimizing sparse matrix dense vector multiplication in the I/O-model. Master’s thesis, Informatik Technische Universität München (2009)Google Scholar
  8. 8.
    Roos, F.F., Jacob, R., Grossmann, J., Fischer, B., Buhmann, J.M., Gruissem, W., Baginsky, S., Widmayer, P.: Pepsplice: cache-efficient search algorithms for comprehensive identification of tandem mass spectra. Bioinformatics 23(22), 3016–3023 (2007)CrossRefGoogle Scholar
  9. 9.
    Vuduc, R.W.: Automatic Performance Tuning of Sparse Matrix Kernels. PhD thesis, University of California, Berkeley (Fall 2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gero Greiner
    • 1
  • Riko Jacob
    • 1
  1. 1.Technische Universität München 

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