What is the largest such that if you can multiply matrices using multiplications (not assuming commutativity of multiplication), then you can multiply matrices in time ? What would the running time of this algorithm be?

Strassensâ€™s algorithm partitions the matrices into 2 matrices, i.e. it divides the problem into sub-problems of size . But the algorithm in question asks for sub-problems of size and in each recursive step it performs matrix multiplications.

Hence, we can write the following recurrence for the running time:

Using case 1 of the Master theorem, the solution of this recurrence is .

For to be , must be smaller than .

Hence, the largest possible is 21.

Running time of this algorithm would be .

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