## 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 | American 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

Funding Information:Research is supported by grants 1878/14, and 1901/14 from the Israel Science Foundation (founded by the Israel Academy of Sciences

Funding Information:

and Humanities) and grant 3-10891 from the Ministry of Science and Technology, Israel. Research is also supported by the Einstein Foundation and the Minerva Foundation. Œis work was supported by the PetaCloud industry-academia consortium. Œis research was supported by a grant from the United States-Israel Bi-national Science Foundation (BSF), Jerusalem, Israel. Œis work was supported by the HUJI Cyber Security Research Center in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Oce. We acknowledge PRACE for awarding us access to Hazel Hen at GCS@HLRS, Germany.

Publisher Copyright:

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

## Keywords

- Bilinear algorithms
- Fast matrix multiplication