Coppersmith–Winograd algorithm

In the mathematical discipline of linear algebra, the Coppersmith–Winograd algorithm is the fastest currently known (2007) algorithm for square matrix multiplication. 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 building block in other algorithms to prove theoretical time bounds. However, unlike the Strassen algorithm, it is not used in practice due to huge constants hidden in the Big O notation.

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 exponent of matrix multiplication is 2, as has long been suspected. It has also been conjectured that no fastest algorithm for matrix multiplication exists, in light of the nearly 20 successive improvements leading to the Coppersmith-Winograd algorithm. A 1982 paper by Coppersmith and Winograd proved that there is no fastest algorithm among Strassen-type bilinear algorithms.

In the group-theoretic approach outlined by Cohn, Umans, et. al., there exists a concrete way of proving estimates of the exponent ${\displaystyle \omega }$ of matrix multiplication via a concept known as the simultaneous triple product property (STPP). To be more specific, the STPP describes the property of a finite group simultaneously "realizing" several independent matrix multiplications via a corresponding family of "index triples" of subsets of the group in such a way that the complexity (rank) of these several multiplications does not exceed the complexity (rank) of the algebra. This leads to general estimates for ${\displaystyle \omega }$ in terms of the the size of the group, the number of STPP triples realized by the group, and the sizes of the components of these triples. The best groups for achieving tight bounds for ${\displaystyle \omega }$ in this way appear to be wreath products of Abelian with symmetric groups. For such wreath products, the choice of appropriate STPP triples in an Abelian group and permutations in a corresponding symmetric group might yield concrete estimates of ${\displaystyle \omega }$ close to 2, as described in Sandeep Murthy.

References

• Sandeep Murthy. The Simultaneous Triple Product Property and Group-theoretic Results for the Exponent of Matrix Multiplication. arXiv:0703145 {{arxiv}}: Check arxiv value (help). 3 April 2007.
• 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.
• Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9:251–280, 1990.