Homework Problem V Solution
CISC 621 – Fall 2003
Problem A (CLRS 28.2-2)
How would you modify Strassen’s algorithm to multiply n x n matrices in which n is not
an exact power of 2? Show that the resulting algorithm runs in time
Θ
(nlg7).
Strassen’s algorithm can be applied to n x n matrix multiplications where n is not an
exact power of 2 by padding the operands with 0’s. Let m = 2k such that 2k - 1 < n < 2k (m
equals 2lgn). Create m x m matrices A’ and B’ by padding A and B respectively.
Applying Strassen’s algorithm, the resulting matrices C’, A’ and B’ appear as follows,
where C’ is the matrix product of A’ and B’:
To obtain the product, we simply extract the matrix C from C’.
The runtime for this method is Θ(mlg7). Since 2k – 1 < n, it follows that m < 2n. Therefore,
the runtime becomes Θ((2n)lg7) = Θ(2 lg7 ⋅ n lg7) = Θ(n lg7).
Problem B (CLRS 28.2-3)
What is the largest k such that if you can multiply 3 x 3 matrices using k multiplications,
then you can multiply n x n matrices in time o(nlg7)? What would the running time of this
algorithm be?
Strassen’s algorithm takes the approach of a recursive multiply with a base condition of
2x2 matrices. We are asked to apply an algorithm using a base condition of a 3x3 matrix
and told it will take k multiplications.
Consider the comparative recursions:
Strassen: T(n) = 7T(n/2) + Θ(n2)
3x3: T(n) = kT(n/3) + Θ(n2)
As the hint points out, case 1 of the Master Theorem applies and the recursive term
dominates. Concentrating on the 3x3 recursion, we want to solve for k such that the
number of multiplies will be less than nlg7. We do so as follows:
Θ(nlg7) ≥ Θ(nlog[3]k), so
