Communication costs of Strassen's matrix multiplication

Communications of the ACM

Volumn 57, Issue 2, 2014, Pages 107-114

  Ballard, Grey ^a   Demmel, James ^a   Holtz, Olga ^a,b   Schwartz, Oded ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ARITHMETIC OPERATIONS; COMMUNICATION COST; FAST MATRIX MULTIPLICATION; FINDING ALGORITHM; MATRIX MULTIPLICATION; MATRIX MULTIPLICATION ALGORITHM; SEQUENTIAL ALGORITHM; STRASSEN'S ALGORITHM;

ALGORITHMS; COMMUNICATION; COSTS; MATRIX ALGEBRA; OPTIMIZATION; PARALLEL ARCHITECTURES;

COST BENEFIT ANALYSIS;

EID: 84893396206     PISSN: 00010782     EISSN: 15577317     Source Type: Journal    
DOI: 10.1145/2556647.2556660     Document Type: Article

References (25)

1. Agarwal, R.C., Balle, S.M., Gustavson, F.G., Joshi, M., Palkar, P. A threedimensional approach to parallel matrix multiplication. IBM J. Res. Dev. 39, 5 (1995), 575-582.
2. Alon, N., Schwartz, O., Shapira, A. An elementary construction of constantdegree expanders. Combinator. Probab. Comput. 17, 3 (2008), 319-327.
3. A nderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S., Sorensen, D. LAPACK's User's Guide, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992. Also available from http://www. netlib.org/lapack/.
4. Ballard, G., Buluç, A., Demmel, J., Grigori, L., Lipshitz, B., Schwartz, O., Toledo, S. Communication Optimal Parallel Multiplication of Sparse Random Matrices. In Proceedings of the 25th ACM Symposium on Parallelism in Algorithms and Architectures, (2013), ACM, New York, NY, USA.
5. Ballard, G., Demmel, J., Holtz, O., Lipshitz, B., Schwartz, O. Brief announcement: Strong scaling of matrix multiplication algorithms and memory-independent communication lower bounds. In Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, (2012), ACM, New York, NY, USA, 77-79.
6. Ballard, G., Demmel, J., Holtz, O., Lipshitz, B., Schwartz, O. Communicationoptimal parallel algorithm for Strassen's matrix multiplication. In Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '12 (2012), ACM, New York, NY, USA, 193-204.
7. Ballard, G., Demmel, J., Holtz, O., Lipshitz, B., Schwartz, O. Graph expansion analysis for communication costs of fast rectangular matrix multiplication. In Design and Analysis of Algorithms. G. Even and D. Rawitz, eds., Volume 7659 of Lecture Notes in Computer Science (2012), Springer, Berlin-Heidelberg, 13-36.
8. Ballard, G., Demmel, J., Holtz, O., Schwartz, O. Graph expansion and communication costs of fast matrix multiplication. In Proceedings of the 23rd Annual ACM Symposium on Parallel Algorithms and Architectures (2011), ACM, New York, NY, USA, 1-12.
9. Ballard, G., Demmel, J., Holtz, O., Schwartz, O. Minimizing communication in numerical linear algebra. SIAM J. Matrix Anal. Appl. 32, 3 (2011), 866-901.
10. Ballard, G., Demmel, J., Holtz, O., Schwartz, O. Graph expansion and communication costs of fast matrix multiplication. J. ACM (Dec. 2012) 59, 6, 32:1-32:23.
11. Cannon, L. A cellular computer to implement the Kalman filter algorithm. PhD thesis, Montana State University, Bozeman, MN (1969).
12. Christ, M., Demmel, J., Knight, N., Scanlon, T., Yelick, K. Communication lower bounds and optimal algorithms for programs that reference arrays - Part I. Manuscript, 2013.
13. Demmel, J., Dumitriu, I., Holtz, O. Fast linear algebra is stable. Numer. Math. 108, 1 (2007), 59-91.
14. Demmel, J., Eliahu, D., Fox, A., Kamil, S., Lipshitz, B., Schwartz, O., Spillinger, O. Communication-optimal parallel recursive rectangular matrix multiplication. In Proceedings of the 27th IEEE International Parallel & Distributed Processing Symposium (IPDPS) (2013), IEEE.
15. Demmel, J., Gearhart, A., Lipshitz, B., Schwartz, O. Perfect strong scaling using no additional energy. In Proceedings of the 27th IEEE International Parallel & Distributed Processing Symposium, IPDPS '13 (2013), IEEE.
16. Fuller, S.H., Millett, L.I., eds. The Future of Computing Performance: Game Over or Next Level? The National Academies Press, Washington, D.C., 2011, 200 pages, http://www.nap.edu.
17. Graham, S.L., Snir, M., Patterson, C.A., eds. Getting up to Speed: The Future of Supercomputing. Report of National Research Council of the National Academies Sciences. The National Academies Press, Washington, D.C., 2004, 289 pages, http://www.nap.edu.
18. Hong, J.W., Kung, H.T. I/O complexity: The red-blue pebble game. In STOC '81: Proceedings of the 13th annual ACM Symposium on Theory of Computing (1981), ACM, New York, NY, USA, 326-333.
19. Hoory, S., Linial, N., Wigderson, A. Expander graphs and their applications. Bull. AMS 43(4), (2006), 439-561.
20. I rony, D., Toledo, S., Tiskin, A. Communication lower bounds for distributed-memory matrix multiplication. J. Parallel Distrib. Comput. 64, 9, (2004), 1017-1026.
21. Lipshitz, B., Ballard, G., Demmel, J., Schwartz, O. Communication- avoiding parallel Strassen: Implementation and performance. In Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis, (2012), IEEE Computer Society Press, Los Alamitos, CA, USA, 101:1-101:11.
22. McColl, W.F., Tiskin, A. Memoryefficient matrix multiplication in the BSP model. Algorithmica 24 (1999), 287-297.
23. Solomonik, E., Demmel, J. Communication- optimal parallel 2.5D matrix multiplication and LU factorization algorithms. In Proceedings of the 17th International European Conference on Parallel and Distributed Computing (2011), Springer.
24. Strassen, V. Gaussian elimination is not optimal. Numer. Math. 13 (1969), 354-356.
25. Williams, V.V. Multiplying matrices faster than Coppersmith-Winograd. In Proceedings of the 44th Symposium on Theory of Computing, STO C '12 (2012), ACM, New York, NY, USA, 887-898.