Information AboutCoppersmith-winograd Algorithm |
| CATEGORIES ABOUT COPPERSMITH–WINOGRAD ALGORITHM | |
| numerical linear algebra | |
| matrix theory | |
| algorithms | |
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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-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 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 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 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 close to 2, as described in Sandeep Murthy . REFERENCES
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