## Abstract

Strassen's algorithm (1969) was the first sub-cubic matrix multiplication algorithm. Winograd (1971) improved its complexity by a constant factor. Many asymptotic improvements followed. Unfortunately, most of them have done so at the cost of very large, o.en gigantic, hidden constants. Consequently, Strassen-Winograd's O (nlog2^{7}) algorithm o.en outperforms other matrix multiplication algorithms for all feasible matrix dimensions. The leading coefficient of Strassen-Winograd's algorithm was believed to be optimal for matrix multiplication algorithms with 2 × 2 base case, due to a lower bound of Probert (1976). Surprisingly, we obtain a faster matrix multiplication algorithm, with the same base case size and asymptotic complexity as Strassen- Winograd's algorithm, but with the coefficient reduced from 6 to 5. To this end, we extend Bodrato's (2010) method for matrix squaring, and transform matrices to an alternative basis. We prove a generalization of Probert's lower bound that holds under change of basis, showing that for matrix multiplication algorithms with a 2×2 base case, the leading coefficient of our algorithm cannot be further reduced, hence optimal. We apply our technique to other Strassenlike algorithms, improving their arithmetic and communication costs by significant constant factors.

Original language | English |
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Title of host publication | SPAA 2017 - Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures |

Publisher | Association for Computing Machinery |

Pages | 101-110 |

Number of pages | 10 |

ISBN (Electronic) | 9781450345934 |

DOIs | |

State | Published - 24 Jul 2017 |

Event | 29th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2017 - Washington, United States Duration: 24 Jul 2017 → 26 Jul 2017 |

### Publication series

Name | Annual ACM Symposium on Parallelism in Algorithms and Architectures |
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Volume | Part F129316 |

### Conference

Conference | 29th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2017 |
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Country/Territory | United States |

City | Washington |

Period | 24/07/17 → 26/07/17 |

### Bibliographical note

Publisher Copyright:© 2017 Copyright held by the owner/author(s).

## Keywords

- Bilinear algorithms
- Fast matrix multiplication