Coppersmith–Winograd algorithm: Difference between revisions

Revision as of 06:32, 15 December 2007

Revision as of 18:37, 24 December 2007

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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 theory|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 ([[O-speedup]]).

In the group-theoretic approach outlined by Cohn, Umans, et al., there exists a concrete way of proving estimates of the exponent <math>\omega</math> 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 <math>\omega</math> in terms of 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 <math>\omega</math> 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 <math>\omega</math> 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|archive=cs.CS|id=0703145}}. 3 April 2007.

* Henry Cohn, Robert Kleinberg, Balazs Szegedy, and Chris Umans. Group-theoretic Algorithms for Matrix Multiplication. {{arXiv|archive=math.GR|id=0511460}}. ''Proceedings of the 46th Annual Symposium on Foundations of Computer Science'', 23-25 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379&ndash;388.

* [[Don Coppersmith]] and [[Shmuel Winograd]]. Matrix multiplication via arithmetic progressions. ''Journal of Symbolic Computation'', 9:251&ndash;280, 1990.

* Henry Cohn, Robert Kleinberg, Balazs Szegedy, and Chris Umans. Group-theoretic Algorithms for Matrix Multiplication. {{arXiv|archive=math.GR|id=0511460}}. ''Proceedings of the 46th Annual Symposium on Foundations of Computer Science'', 23-25 October 2005, Pittsburgh, PA, IEEE Computer Society, pp. 379&ndash;388.

[[Category:Numerical linear algebra]]