## Journal of the ACM (JACM)

##### NEWS

The Journal of the ACM (JACM) provides coverage of the most significant work on principles of computer science, broadly construed. The scope of research we cover encompasses contributions of lasting value to any area of computer science. To be accepted, a paper must be judged to be truly outstanding in its field.  JACM is interested  in work in core computer science and at the boundaries, both the boundaries of subdisciplines of computer science and the boundaries between computer science and other fields.  READ MORE

### Editorial Process

The Journal of the ACM begins the refereeing process with a "quick review", to check whether the manuscript has a plausible chance of meeting JACM's high standards, even if all the claimed results are correct. JACM tries to cover a broad spectrum of areas, and can only accept 4-5 papers in any given area every year. Thus, we try to focus on the most significant papers in each area, that would be of interest to the broad community, and reject many papers that would be accepted by other journals. READ MORE

### Important Note on P/NP

Some submissions purport to solve a long-standing open problem in complexity theory, such as the P/NP problem. Many of these turn out to be mistaken, and such submissions tax JACM volunteer editors and reviewers.  READ MORE

Strassen's algorithm (1969) was the first sub-cubic matrix multiplication algorithm. Winograd (1971) improved the leading coefficient of its complexity from 6 to 7. Many asymptotic improvements followed. Unfortunately, most of them have done so at the cost of very large, often gigantic, hidden constants. Consequently, Strassen-Winograd's $O\left(n^{\log_{2}7}\right)$ algorithm often outperforms other fast 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 $2\times2$ base case, due to a lower bound by 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 leading 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 $2\times2$ base case, the leading coefficient of our algorithm cannot be further reduced, hence optimal. We apply our method to other fast matrix multiplication algorithms, improving their arithmetic and communication costs by significant constant factors.