# 如何实现Strassen的矩阵乘法算法？

2021年3月18日15:34:37 发表评论 693 次浏览

Strassen的矩阵乘法方法是一种典型的分而治之算法。我们已经讨论了Strassen的算法这里。但是, 让我们再次了解分而治之方法背后的实质并加以实施。

``````// CPP program to implement Strassen’s Matrix
// Multiplication Algorithm
#include <bits/stdc++.h>
using namespace std;
typedef long long lld;

/* Strassen's Algorithm for matrix multiplication
Complexity:    O(n^2.808) */

inline lld** MatrixMultiply(lld** a, lld** b, int n, int l, int m)
{
lld** c = new lld*[n];
for ( int i = 0; i < n; i++)
c[i] = new lld[m];

for ( int i = 0; i < n; i++) {
for ( int j = 0; j < m; j++) {
c[i][j] = 0;
for ( int k = 0; k < l; k++) {
c[i][j] += a[i][k] * b[k][j];
}
}
}
return c;
}

inline lld** Strassen(lld** a, lld** b, int n, int l, int m)
{
if (n == 1 || l == 1 || m == 1)
return MatrixMultiply(a, b, n, l, m);

lld** c = new lld*[n];
for ( int i = 0; i < n; i++)
c[i] = new lld[m];

int adjN = (n >> 1) + (n & 1);
int adjL = (l >> 1) + (l & 1);
int adjM = (m >> 1) + (m & 1);

lld**** As = new lld***[2];
for ( int x = 0; x < 2; x++) {
As[x] = new lld**[2];
for ( int y = 0; y < 2; y++) {
for ( int i = 0; i < adjN; i++) {
for ( int j = 0; j < adjL; j++) {
int I = i + (x & 1) * adjN;
int J = j + (y & 1) * adjL;
As[x][y][i][j] = (I < n && J < l) ? a[I][J] : 0;
}
}
}
}

lld**** Bs = new lld***[2];
for ( int x = 0; x < 2; x++) {
Bs[x] = new lld**[2];
for ( int y = 0; y < 2; y++) {
for ( int i = 0; i < adjL; i++) {
for ( int j = 0; j < adjM; j++) {
int I = i + (x & 1) * adjL;
int J = j + (y & 1) * adjM;
Bs[x][y][i][j] = (I < l && J < m) ? b[I][J] : 0;
}
}
}
}

lld*** s = new lld**[10];
for ( int i = 0; i < 10; i++) {
switch (i) {
case 0:
for ( int j = 0; j < adjL; j++) {
for ( int k = 0; k < adjM; k++) {
s[i][j][k] = Bs[0][1][j][k] - Bs[1][1][j][k];
}
}
break ;
case 1:
for ( int j = 0; j < adjN; j++) {
for ( int k = 0; k < adjL; k++) {
s[i][j][k] = As[0][0][j][k] + As[0][1][j][k];
}
}
break ;
case 2:
for ( int j = 0; j < adjN; j++) {
for ( int k = 0; k < adjL; k++) {
s[i][j][k] = As[1][0][j][k] + As[1][1][j][k];
}
}
break ;
case 3:
for ( int j = 0; j < adjL; j++) {
for ( int k = 0; k < adjM; k++) {
s[i][j][k] = Bs[1][0][j][k] - Bs[0][0][j][k];
}
}
break ;
case 4:
for ( int j = 0; j < adjN; j++) {
for ( int k = 0; k < adjL; k++) {
s[i][j][k] = As[0][0][j][k] + As[1][1][j][k];
}
}
break ;
case 5:
for ( int j = 0; j < adjL; j++) {
for ( int k = 0; k < adjM; k++) {
s[i][j][k] = Bs[0][0][j][k] + Bs[1][1][j][k];
}
}
break ;
case 6:
for ( int j = 0; j < adjN; j++) {
for ( int k = 0; k < adjL; k++) {
s[i][j][k] = As[0][1][j][k] - As[1][1][j][k];
}
}
break ;
case 7:
for ( int j = 0; j < adjL; j++) {
for ( int k = 0; k < adjM; k++) {
s[i][j][k] = Bs[1][0][j][k] + Bs[1][1][j][k];
}
}
break ;
case 8:
for ( int j = 0; j < adjN; j++) {
for ( int k = 0; k < adjL; k++) {
s[i][j][k] = As[0][0][j][k] - As[1][0][j][k];
}
}
break ;
case 9:
for ( int j = 0; j < adjL; j++) {
for ( int k = 0; k < adjM; k++) {
s[i][j][k] = Bs[0][0][j][k] + Bs[0][1][j][k];
}
}
break ;
}
}

lld*** p = new lld**[7];

for ( int i = 0; i < adjN; i++) {
for ( int j = 0; j < adjM; j++) {
c[i][j] = p[4][i][j] + p[3][i][j] - p[1][i][j] + p[5][i][j];
if (j + adjM < m)
c[i][j + adjM] = p[0][i][j] + p[1][i][j];
if (i + adjN < n)
c[i + adjN][j] = p[2][i][j] + p[3][i][j];
if (i + adjN < n && j + adjM < m)
c[i + adjN][j + adjM] = p[4][i][j] + p[0][i][j] - p[2][i][j] - p[6][i][j];
}
}

for ( int x = 0; x < 2; x++) {
for ( int y = 0; y < 2; y++) {
for ( int i = 0; i < adjN; i++) {
delete [] As[x][y][i];
}
delete [] As[x][y];
}
delete [] As[x];
}
delete [] As;

for ( int x = 0; x < 2; x++) {
for ( int y = 0; y < 2; y++) {
for ( int i = 0; i < adjL; i++) {
delete [] Bs[x][y][i];
}
delete [] Bs[x][y];
}
delete [] Bs[x];
}
delete [] Bs;

for ( int i = 0; i < 10; i++) {
switch (i) {
case 0:
case 3:
case 5:
case 7:
case 9:
for ( int j = 0; j < adjL; j++) {
delete [] s[i][j];
}
break ;
case 1:
case 2:
case 4:
case 6:
case 8:
for ( int j = 0; j < adjN; j++) {
delete [] s[i][j];
}
break ;
}
delete [] s[i];
}
delete [] s;

for ( int i = 0; i < 7; i++) {
for ( int j = 0; j < (n >> 1); j++) {
delete [] p[i][j];
}
delete [] p[i];
}
delete [] p;

return c;
}

int main()
{
lld** matA;
matA = new lld*[2];
for ( int i = 0; i < 2; i++)
matA[i] = new lld[3];
matA[0][0] = 1;
matA[0][1] = 2;
matA[0][2] = 3;
matA[1][0] = 4;
matA[1][1] = 5;
matA[1][2] = 6;

lld** matB;
matB = new lld*[3];
for ( int i = 0; i < 3; i++)
matB[i] = new lld[2];
matB[0][0] = 7;
matB[0][1] = 8;
matB[1][0] = 9;
matB[1][1] = 10;
matB[2][0] = 11;
matB[2][1] = 12;

lld** matC = Strassen(matA, matB, 2, 3, 2);
for ( int i = 0; i < 2; i++) {
for ( int j = 0; j < 2; j++) {
printf ( "%lld " , matC[i][j]);
}
printf ( "\n" );
}

return 0;
}``````
``````Output:  58   64
139  154``````