Strassens Matrix Multiplication Sibel KIRMIZIGL Basic Matrix Multiplication

Strassen's Matrix Multiplication Sibel KIRMIZIGÜL

Basic Matrix Multiplication Suppose we want to multiply two matrices of size N x N: for example A x B = C. C 11 = a 11 b 11 + a 12 b 21 C 12 = a 11 b 12 + a 12 b 22 C 21 = a 21 b 11 + a 22 b 21 C 22 = a 21 b 12 + a 22 b 22 2 x 2 matrix multiplication can be accomplished in 8 multiplication. (2 log 28 =23)

Basic Matrix Multiplication void matrix_mult (){ for (i = 1; i <= N; i++) { for (j = 1; j <= N; j++) { compute Ci, j; } }} Time analysis algorithm

Strassens’s Matrix Multiplication Strassen showed that 2 x 2 matrix multiplication can be accomplished in 7 multiplication and 18 additions or subtractions. . (2 log 27 =22. 807) This reduce can be done by Divide and Conquer Approach.

Divide-and-Conquer Divide-and conquer is a general algorithm design paradigm: n n n Divide: divide the input data S in two or more disjoint subsets S 1, S 2, … Recur: solve the subproblems recursively Conquer: combine the solutions for S 1, S 2, …, into a solution for S The base case for the recursion are subproblems of constant size Analysis can be done using recurrence equations

Divide and Conquer Matrix Multiply A A 0 A 1 A 2 A 3 B = B 0 B 1 B 2 B 3 = R A 0 B 0+A 1 B 2 A 0 B 1+A 1 B 3 A 2 B 0+A 3 B 2 A 2 B 1+A 3 B 3 • Divide matrices into sub-matrices: A 0 , A 1, A 2 etc • Use blocked matrix multiply equations • Recursively multiply sub-matrices

Divide and Conquer Matrix Multiply A B = a 0 b 0 = R a 0 b 0 • Terminate recursion with a simple base case

Strassens’s Matrix Multiplication P 1 = (A 11+ A 22)(B 11+B 22) P 2 = (A 21 + A 22) * B 11 P 3 = A 11 * (B 12 - B 22) P 4 = A 22 * (B 21 - B 11) P 5 = (A 11 + A 12) * B 22 P 6 = (A 21 - A 11) * (B 11 + B 12) P 7 = (A 12 - A 22) * (B 21 + B 22) C 11 = P 1 + P 4 - P 5 + P 7 C 12 = P 3 + P 5 C 21 = P 2 + P 4 C 22 = P 1 + P 3 - P 2 + P 6

Comparison C 11 = P 1 + P 4 - P 5 + P 7 = (A 11+ A 22)(B 11+B 22) + A 22 * (B 21 - B 11) - (A 11 + A 12) * B 22+ (A 12 - A 22) * (B 21 + B 22) = A 11 B 11 + A 11 B 22 + A 22 B 11 + A 22 B 22 + A 22 B 21 – A 22 B 11 A 11 B 22 -A 12 B 22 + A 12 B 21 + A 12 B 22 – A 22 B 21 – A 22 B 22 = A 11 B 11 + A 12 B 21

Strassen Algorithm void matmul(int *A, int *B, int *R, int n) { if (n == 1) { (*R) += (*A) * (*B); } else { matmul(A, B, R, n/4); matmul(A, B+(n/4), R+(n/4), n/4); matmul(A+2*(n/4), B, R+2*(n/4), n/4); matmul(A+2*(n/4), B+(n/4), R+3*(n/4), n/4); matmul(A+(n/4), B+2*(n/4), R, n/4); matmul(A+(n/4), B+3*(n/4), R+(n/4), n/4); matmul(A+3*(n/4), B+2*(n/4), R+2*(n/4), n/4); matmul(A+3*(n/4), B+3*(n/4), R+3*(n/4), n/4); } Divide matrices in sub-matrices and recursively multiply sub-matrices

Time Analysis