Today: Matrix Multiplication and Set Equality
We have a nice implementation of a fast matrix multiplication algorithm, say the Coppersmith-Winograd algorithm, which runs in O(n^2.38). Its unbelievably complicated and you’d like to know if it even computes the correct answer.
How could you verify the answer? Well we could just do the vanilla matrix multiplication, but that runs in O(n^3) time, which would defeat the purpose of having a faster algorithm. How do we do better? I have an idea! Let’s try a randomizing algorithm!
Note, that matrix multiplication is associative. One thing we can do is choose a random vector r \in 2^n (flip coins, look at the stars). Compute (A*B)*r = A*(B*r) = A*d = e in O(n^2) time. Likewise, compute C*r = f in O(n^2) time. We can show that if A*B != C, then e != f with probability at least 1/2.
Since this is one of the final exam questions, make sure you understand all of the details!
I’ll be making a separate post for the final exam so that everyone understands what they have to do.