第四章 分治策略 4.2 矩阵乘法的Strassen算法

package chap04_Divide_And_Conquer;

import static org.junit.Assert.*;

import java.util.Arrays;

import org.junit.Test;

/**
 * 矩阵相乘的算法
 *
 * @author xiaojintao
 *
 */
public class MatrixOperation {
    /**
     * 普通的矩阵相乘算法,c=a*b。其中，a、b都是n*n的方阵
     *
     * @param a
     * @param b
     * @return c
     */
    static int[][] matrixMultiplicationByCommonMethod(int[][] a, int[][] b) {
        int n = a.length;
        int[][] c = new int[n][n];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i][j] = 0;
                for (int k = 0; k < n; k++) {
                    c[i][j] += a[i][k] * b[k][j];
                }
            }
        }
        return c;
    }

    /**
     * strassen 算法求矩阵乘法 n为2的幂
     *
     * @param a
     * @param b
     * @return
     */
    static int[][] matrixMultiplicationByStrassen(int[][] a, int[][] b) {
        int n = a.length;
        if (n == 1) {
            int[][] c = new int[1][1];
            c[0][0] = a[0][0] * b[0][0];
            return c;
        }
        int m = n / 2;
        int[][] a11, a12, a21, a22, b11, b12, b21, b22;
        int[][] c = new int[n][n];
        a11 = new int[m][m];
        a12 = new int[m][m];
        a21 = new int[m][m];
        a22 = new int[m][m];
        b11 = new int[m][m];
        b12 = new int[m][m];
        b21 = new int[m][m];
        b22 = new int[m][m];

        for (int i = 0; i < m; i++) {
            for (int j = 0; j < m; j++) {
                a11[i][j] = a[i][j];
            }
        }
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < m; j++) {
                b11[i][j] = b[i][j];
            }
        }
        for (int i = 0; i < m; i++) {
            for (int j = m; j < n; j++) {
                a12[i][j - m] = a[i][j];
            }
        }
        for (int i = 0; i < m; i++) {
            for (int j = m; j < n; j++) {
                b12[i][j - m] = b[i][j];
            }
        }
        for (int i = m; i < n; i++) {
            for (int j = 0; j < m; j++) {
                a21[i - m][j] = a[i][j];
            }
        }
        for (int i = m; i < n; i++) {
            for (int j = 0; j < m; j++) {
                b21[i - m][j] = b[i][j];
            }
        }
        for (int i = m; i < n; i++) {
            for (int j = m; j < n; j++) {
                a22[i - m][j - m] = a[i][j];
            }
        }
        for (int i = m; i < n; i++) {
            for (int j = m; j < n; j++) {
                b22[i - m][j - m] = b[i][j];
            }
        }
        int[][] s1 = matrixMinus(b12, b22);
        int[][] s2 = matrixAdd(a11, a12);
        int[][] s3 = matrixAdd(a21, a22);
        int[][] s4 = matrixMinus(b21, b11);
        int[][] s5 = matrixAdd(a11, a22);
        int[][] s6 = matrixAdd(b11, b22);
        int[][] s7 = matrixMinus(a12, a22);
        int[][] s8 = matrixAdd(b21, b22);
        int[][] s9 = matrixMinus(a11, a21);
        int[][] s10 = matrixAdd(b11, b12);

        int[][] p1 = matrixMultiplicationByStrassen(a11, s1);
        int[][] p2 = matrixMultiplicationByStrassen(s2, b22);
        int[][] p3 = matrixMultiplicationByStrassen(s3, b11);
        int[][] p4 = matrixMultiplicationByStrassen(a22, s4);
        int[][] p5 = matrixMultiplicationByStrassen(s5, s6);
        int[][] p6 = matrixMultiplicationByStrassen(s7, s8);
        int[][] p7 = matrixMultiplicationByStrassen(s9, s10);

        int[][] t1, t2, t3;
        t1 = matrixAdd(p5, p4);
        t2 = matrixMinus(t1, p2);
        int[][] c11 = matrixAdd(t2, p6);
        int[][] c12 = matrixAdd(p1, p2);
        int[][] c21 = matrixAdd(p3, p4);
        t1 = matrixAdd(p5, p1);
        t2 = matrixMinus(t1, p3);
        int[][] c22 = matrixMinus(t2, p7);
        c = matrixConbine(c11, c12, c21, c22);
        return c;
    }

    /**
     * 矩阵加法 c=a+b
     *
     * @param a
     * @param b
     * @return
     */
    static int[][] matrixAdd(int[][] a, int[][] b) {
        int n = a.length;
        int[][] c = new int[n][n];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i][j] = a[i][j] + b[i][j];
            }
        }
        return c;
    }

    /**
     * 矩阵减法 c=a-b
     *
     * @param a
     * @param b
     * @return
     */
    static int[][] matrixMinus(int[][] a, int[][] b) {
        int n = a.length;
        int[][] c = new int[n][n];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i][j] = a[i][j] - b[i][j];
            }
        }
        return c;
    }

    /**
     * 将矩阵的四个部分组合
     *
     * @param t11
     * @param t12
     * @param t21
     * @param t22
     * @return
     */
    protected static int[][] matrixConbine(int[][] t11, int[][] t12,
            int[][] t21, int[][] t22) {
        int n = t11.length;
        int m = 2 * n;
        int[][] c = new int[m][m];
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i][j] = t11[i][j];
            }
        }
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i][j + n] = t12[i][j];
            }
        }
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i + n][j] = t21[i][j];
            }
        }
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < n; j++) {
                c[i + n][j + n] = t22[i][j];
            }
        }
        return c;
    }

    @Test
    public void testName() throws Exception {
        // int[][] a = { { 1, 2, 3 }, { 4, 5, 6 }, { 7, 8, 9 } };
        // int[][] b = { { 1, 3, 5 }, { 2, 4, 6 }, { 9, 8, 7 } };
        // int[][] c = commonMatrixMultiplication(a, b);
        // int[][] c = matrixAdd(a, b);

        int[][] m = { { 1, 2, 3, 4 }, { 5, 6, 7, 8 }, { 9, 10, 11, 12 },
                { 13, 14, 15, 16 } };
        int[][] n = { { 1, 3, 5, 7 }, { 2, 4, 6, 8 }, { 4, 3, 2, 1 },
                { 9, 8, 7, 6 } };

        int[][] c = matrixMultiplicationByStrassen(m, n);
        System.out.println(Arrays.deepToString(c));
        int[][] d = matrixMultiplicationByCommonMethod(m, n);
        System.out.println(Arrays.deepToString(d));
    }
}

暴力求解复杂度为O（n3），Strassen算法为O（n log7）