Posted by: atri | November 29, 2011

## A new faster algorithm for matrix multiplication

Yesterday Virginia Vassilevska Williams announced a new faster algorithm for matrix multiplication: this is the first improvement in more than two decades. (Continuing the tradition of problems resisting  improvements for a while before finally giving way with independent discoveries, Andrew Stothers also had a similar (but a bit weaker result) last year in his PhD thesis). See Scott Aaronson’s blog post and Dick Lipton and (our own) Ken Regan’s blog post for more on the results.

Background on the problem: the problem is, given two $n\times n$ matrices $A$ and $B$ compute their product $A\times B$. The trivial algorithm takes $O(n^3)$ time. The first improvement was in an algorithm due to Strassen who presented a divide and conquer algorithm that runs in time $O(n^{\log_2{7}})$. The “trick” in some sense is similar to the one we saw for integer multiplication: instead of doing the naive number of multiplications over subproblems (in this case $8$), one uses more (matrix) addition and subtraction to get away with a fewer number of sub-problems (in this case $7$). The problem is very fundamental and has tons of applications (see the above papers for more context and history).