Can you beat the Strassen's Algorithm?
His algorithm will use the divide-and-conquer method, dividing each matrix into pieces of size \(n/4 \times n/4\) and the divide and combine steps together will take \( \Theta(n^2) \) time.
If his algorithm creates \(a\) subproblems, then the recurrence for the running time \(T(n)\) becomes \[T(n) = a\,T(n/4) + \Theta(n^2)\]
What is the largest integer value of \(a\) for which Professor Caesar's algorithm would be asymptotically faster than Strassen's algorithm?
If the running time of the Strassen's algorithm is \(T(n)\), it satisfies the relation \[ T(n) = 7\, T(n/2) + \Theta (n^2) \]