Coppersmith–Winograd algorithm

In the mathematical discipline of linear algebra, the Coppersmith–Winograd algorithm, named after Don Coppersmith and Shmuel Winograd, is the asymptotically fastest known algorithm for square matrix multiplication as of 2008. It can multiply two ${\displaystyle n\times n}$ matrices in ${\displaystyle O(n^{2.376})}$ time (see Big O notation). This is an improvement over the trivial ${\displaystyle O(n^{3})}$ time algorithm and the ${\displaystyle O(n^{2.807})}$ time Strassen algorithm. It might be possible to improve the exponent further; however, the exponent must be at least 2 (because an ${\displaystyle n\times n}$ matrix has ${\displaystyle n^{2}}$ values, and all of them have to be read at least once to calculate the exact result).

The Coppersmith–Winograd algorithm is frequently used as a building block in other algorithms to prove theoretical time bounds. However, unlike the Strassen algorithm, it is not used in practice because it only provides an advantage for matrices so large that they cannot be processed by modern hardware.^[1]

Henry Cohn, Robert Kleinberg, Balázs Szegedy and Christopher Umans have rederived the Coppersmith–Winograd algorithm using a group-theoretic construction. They also show that either of two different conjectures would imply that the optimal exponent of matrix multiplication is 2, as has long been suspected.

References

1. Robinson, Sara (2005), "Toward an Optimal Algorithm for Matrix Multiplication" (PDF), SIAM News, 38 (9)

• Henry Cohn, Robert Kleinberg, Balazs Szegedy, and Chris Umans. Group-theoretic Algorithms for Matrix Multiplication. arXiv:0511460 {{arxiv}}: Check arxiv value (help). Proceedings of the 46th Annual Symposium on Foundations of Computer Science, 23–25 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379–388.
• Coppersmith, Don; Winograd, Shmuel (1990), "Matrix multiplication via arithmetic progressions" (PDF), Journal of Symbolic Computation, 9 (3): 251–280, doi:10.1016/S0747-7171(08)80013-2.