Professor Caesar wishes to develop a matrix-multiplication algorithm that is asymptotically faster than 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. He needs to determine how many subproblems his algorithm has to create in order to beat Strassen’s algorithm. If his algorithm creates \(a\) subproblems, then the recurrence for the running time \(T(n)\) becomes \(T(n) = aT(n/4) + \Theta(n^2)\). What is the largest integer value of \(a\) for which Professor Caesar’s algorithm would be asymptotically faster than Strassen’s algorithm?
Assymptotic running time for Strassen’s algorithm is \(S(n) = \Theta(n^{\lg 7})\)
Now, when \(a\) increases, number of subproblems determines the assymptotic running time of the problem and case 1 of master theorem applies. So, in worst case, assymptotic running time of the algortihm will be \(T(n) = \Theta(n^{\log_b a}) = \Theta(n^{\log_4 a}) = \Theta(n^{\log_2 \sqrt a})\)
Now, for \(T(n)\) to be smaller than \(S(n)\), \(n^{\lg \sqrt a}\) must be smaller than \(n^{\lg 7}\).
\[\begin {aligned} n^{\lg \sqrt a} & < n^{\lg 7} \\ \lg \sqrt a & < \lg 7 \\ \sqrt a & < 7 \\ a & < 49 \\ \end {aligned}\]Hence, largest integer value of \(a\) is \(48\).