Coppersmith–Winograd algorithm

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In the mathematical discipline of linear algebra, the Coppersmith–Winograd algorithm, named after Don Coppersmith and Shmuel Winograd, is the fastest currently known (2008) algorithm for square matrix multiplication. It can multiply two n \times n matrices in O(n^{2.376}) \!\ time (see Big O notation). This is an improvement over the trivial O(n^3) \!\ time algorithm and the 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 n \times n matrix has n2 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.

[edit] References

  • Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. Journal of Symbolic Computation, 9:251–280, 1990.
  • Henry Cohn, Robert Kleinberg, Balazs Szegedy, and Chris Umans. Group-theoretic Algorithms for Matrix Multiplication. arXiv:math.GR/0511460. Proceedings of the 46th Annual Symposium on Foundations of Computer Science, 23-25 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379–388.