ECCC-Report TR17-131https://eccc.weizmann.ac.il/report/2017/131Comments and Revisions published for TR17-131en-usFri, 01 Sep 2017 19:41:37 +0300
Paper TR17-131
| Designing Strassen's algorithm |
Joshua Grochow,
Cris Moore
https://eccc.weizmann.ac.il/report/2017/131In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x 2 matrix multiplication could be performed with only 7 multiplications instead of 8. The latter construction was arrived at by a process of elimination and appears to come out of thin air. Here, we give the simplest and most transparent proof of Strassen's algorithm that we are aware of, using only a simple unitary 2-design and a few easy lines of calculation. Moreover, using basic facts from the representation theory of finite groups, we use 2-designs coming from group orbits to generalize our construction to all n (although the resulting algorithms aren't optimal for n at least 3).Fri, 01 Sep 2017 19:41:37 +0300https://eccc.weizmann.ac.il/report/2017/131