实施 Strassen 矩阵乘法算法的问题

Posted 2023-02-22

【中文标题】实施 Strassen 矩阵乘法算法的问题

【英文标题】：Trouble With Implementing Strassen's Algorithm for Matrix Multiplication

【发布时间】：2017-07-12 00:44:59

【问题描述】：

在过去的几个小时里，我一直在尝试实现 Strassen 的矩阵乘法算法，但无法获得正确的乘积。我认为我的辅助函数之一（helpSub、createProd、helpProduct）可能是我的 strass2 函数的问题或格式（命令顺序等）。任何提示都会受到欢迎，因为我完全被难住了。我一直在使用两个 4 x 4 矩阵作为测试矩阵。我已经尝试了很多我在互联网上看到的 p1-p7 和 c1-c4 变体，但似乎都没有。下面是我创建的类。

 /* @author williamnewman

public class strassen2 

//Main Strassen multiplication function
//BASE CASE:
int [][] strass2(int[][] x, int[][]y)
    if(x.length == 1 && y.length == 1)
        System.out.println("Donezo");
        int [][] nu = new int[1][1];
        nu[0][0] = x[0][0] * y[0][0];
        return nu;

    
    else
   int[][] a,b,c,d,e,f,g,h;
   int dim = x.length/2;

//Dividing two matrices into 8 sub matrices
  System.out.println("A<B<C");
   a = helpSub(0,0,x);
   C(a);
   b = helpSub(0,dim,x);

   C(b);
   c = helpSub(dim,0,x);
   C(c);
   d = helpSub(dim,dim,x);
   C(d);
   e = helpSub(0,0,y);
   C(e);
   f = helpSub(0,dim,y);
   C(f);
   g = helpSub(dim,0,y);
   C(g);
   h = helpSub(dim,dim,y);
   C(h);

   int[][] p1,p2,p3,p4,p5,p6,p7;


//Creating p1-p7
   /
   p1 = strass2(a,subtract(f,h));
   p2 = strass2(h, add(a,b));
   p3 = strass2(e,add(c,d));
   p4 = strass2(d,subtract(g,e));
   p5 = strass2(add(a,d),add(e,h));
   p6 = strass2(subtract(b,d),add(g,h));
   p7 = strass2(subtract(a,c),add(e,f));
   int [][] prod;
   int [][] c1,c2,c3,c4;

//Creating c1-c4
   c1 = subtract(add(p6,p5),subtract(p4,p2));
   c2 = add(p1,p2);
   c3 = add(p3,p4);
   c4 = subtract(add(p1,p5),subtract(p3,p7));
   C(c1);
   System.out.println("C1::");
   C(c2);
   System.out.println("C2::");
   C(c3);
   System.out.println("C3::");
   C(c4);
   System.out.println("C4::");
//CREATES PRODUCT MATRIX
   prod = createProd(c1,c2,c3,c4);
   return prod;

    






//Creates product matrix from c1-c4
int[][] createProd(int[][] c1, int[][] c2, int[][] c3, int[][] c4)
    int[][] product = new int[c1.length*2][c1.length*2];
    int mid = c1.length;
    int fin = c1.length * 2;
    helpProduct(0,0,mid,mid,product,c1);
    helpProduct(0,mid,mid,fin,product,c2);
    helpProduct(mid,0,fin,mid,product,c3);
    helpProduct(mid,mid,fin,fin,product,c4);

     System.out.println();
    System.out.println("PRODUCT::!:");
    C(product);
    return product;





    //Helper function to create larger matrix from submatrices
void helpProduct(int x, int y, int z1, int z2,int[][] product, int[][] a1)
    int indR = 0;
    int indC = 0;
    for(int i = x; i < z1; i++)
        indC = 0;
        for(int j = y; j < z2; j++)
            product[i][j] = a1[indR][indC];
            indC++;
        
        indR++;
    



    int[][] helpSub(int x, int y, int[][] mat)
    int[][] sub = new int[mat.length/2][mat.length/2];
    for(int i1 = 0, i2=x; i1 < (mat.length/2); i1++, i2++)
    for(int j1 = 0, j2=y; j1<(mat.length/2); j1++, j2++)
    
            sub[i1][j1] = mat[i2][j2];
                           // System.out.println(sub[i1][j1]);
    
    return sub;




//Normal Matrix Multiplication Function
int[][] multiply(int[][]a,int[][]b)
    MM nu = new MM(a,b);
    return nu.product;


    //Adds one matrix to the next
int[][] add(int[][]a, int[][]b)
    int [][] nu = new int[a.length][a[0].length];
    for(int i = 0; i < a.length; i++)
        for(int j = 0; j < a[i].length;j++)
            nu[i][j] = a[i][j] + b[i][j];
        
    
    return nu;


//Subtracts second matrix from the first
int[][] subtract(int[][] a, int[][] b)
    int [][] sub = new int[a.length][a.length];
    //System.out.println("made it");
    for(int i = 0; i < a.length; i++)
        for(int j = 0; j < a[i].length;j++)
            sub[i][j] = a[i][j] - b[i][j];
        
    
    return sub;

//Prints the matrix
 void C(int[][] product)
    for(int i = 0; i <product.length; i++)
        for(int j = 0; j < product[i].length; j++)
            System.out.print(product[j][i]  + " ");

        
        System.out.println();
    

如果有任何令人困惑的地方，请告诉我，我会更新问题！

这里是主要功能::

      public static void main(String[] args) 
        int [][]a = 1,2,3,4,
            4,3,2,1,
            1,2,3,4,
            4,3,2,1;

        int [][]b = 3,4,5,6,
            3,4,5,6,
            5,4,3,2,
            5,4,3,2
        ;
        MM a1 = new MM(a,b);
        a1.C();
        int[][] prod;
        System.out.println("----");
        strassen2 a2 = new strassen2();
        prod = a2.strass2(a,b);
        a2.C(prod);
    

这是目前的结果（预期结果是显示的第一个 4x4 矩阵，实际结果是显示的最后一个 4x4 矩阵）：

EXPECTED:

44 40 36 32 
36 40 44 48 
44 40 36 32 
36 40 44 48 
----


ACTUAL::
70 78 50 42 
86 86 34 34 
30 38 30 38 
38 54 38 54 

我很确定我的 helpSub() 函数可以正常工作，因为它们生成了正确的 a-h。但是，我在 strass2 递归调用中使用的参数可能存在问题。很抱歉，如果它不够具体，我只是有点精疲力尽，并且很好奇是否有人看到任何明显的问题。

【问题讨论】：

您可能想提及究竟是什么 不起作用。当然，您有某种测试，例如，将两个 4x4 统一矩阵相乘不会给您一个统一矩阵，或其他任何东西。如果是这样，请分享它，使您的示例完整且可运行，提及输入、预期结果以及您得到的错误结果。您还期望如何获得帮助，即有人完整阅读您的代码，甚至没有尝试运行它？这是一种更难的方法，它并不是有史以来最易读的。

【参考方案1】：

抱歉含糊不清，但似乎我已经解决了这个问题。我使用了这个网站上的公式来计算 p1-p7 和 c1-c4。 ([施特拉森矩阵乘法的公式][1]

[1]:http://www.stoimen.com/blog/2012/11/26/computer-algorithms-strassens-matrix-multiplication/)

实施这些公式后，乘积矩阵几乎是正确的，但 4 或值不正确。然后我将基本情况更改为当 x 和 y 的长度等于 2 时，这似乎纠正了 4 个关闭的值。对于那些好奇的人，这里是我为 strassen2 类修改的代码。

/*
 * To change this license header, choose License Headers in Project Properties.
 * To change this template file, choose Tools | Templates
 * and open the template in the editor.
 */
package pkg2a;

/**
 *
 * @author williamnewman
 */
public class strassen2 

    int [][] strass2(int[][] x, int[][]y)
        if(x.length <= 2 && y.length <= 2) //!!!! MODIFICATION HERE !!
            return multiply(x,y);

        
        else
       int[][] a,b,c,d,e,f,g,h;
       int dim = x.length/2;

      System.out.println("A<B<C");
       a = helpSub(0,0,x);
       //C(a);
       b = helpSub(0,dim,x);

       //C(b);
       c = helpSub(dim,0,x);
       //C(c);
       d = helpSub(dim,dim,x);
       //C(d);
       e = helpSub(0,0,y);
       //C(e);
       f = helpSub(0,dim,y);
       //C(f);
       g = helpSub(dim,0,y);
       //C(g);
       h = helpSub(dim,dim,y);
       //C(h);

       int[][] p1,p2,p3,p4,p5,p6,p7;
      // createSub(x,y,a,b,c,d,e,f,g,h);
      int[] s1,s2,s3,s4,s5,s6,s7,s8,s9,s10; 

      //MODIFICATION HERE
       p1 = strass2(a,subtract(f,h));
       p2 = strass2(add(a,b),h);
       p3 = strass2(add(c,d),e);
       p4 = strass2(d,subtract(g,e));
       p5 = strass2(add(a,d),add(e,h));
       p6 = strass2(subtract(b,d),add(g,h));
       p7 = strass2(subtract(a,c),add(e,f));
       int [][] prod;
       int [][] c1,c2,c3,c4;
       c1 = subtract(add(p5,p4),subtract(p2,p6));
       c2 = add(p1,p2);
       c3 = add(p3,p4);
       c4 = subtract(add(p1,p5),add(p3,p7));
       //C(c1);
       //System.out.println("C1::");
       //C(c2);
       //System.out.println("C2::");
       //C(c3);
       //System.out.println("C3::");
       //C(c4);
       //System.out.println("C4::");
       prod = createProd(c1,c2,c3,c4);
       return prod;

        




    

    int[][] createProd(int[][] c1, int[][] c2, int[][] c3, int[][] c4)
        int[][] product = new int[c1.length*2][c1.length*2];
        int mid = c1.length;
        int fin = c1.length * 2;
        helpProduct(0,0,mid,mid,product,c1);
        helpProduct(0,mid,mid,fin,product,c2);
        helpProduct(mid,0,fin,mid,product,c3);
        helpProduct(mid,mid,fin,fin,product,c4);

         System.out.println();
        System.out.println("PRODUCT::!:");
        //C(product);
        return product;



    

        //Helper function to create larger matrix from submatrices
    void helpProduct(int x, int y, int z1, int z2,int[][] product, int[][] a1)
        int indR = 0;
        int indC = 0;
        for(int i = x; i < z1; i++)
            indC = 0;
            for(int j = y; j < z2; j++)
                product[i][j] = a1[indR][indC];
                indC++;
            
            indR++;
        
    

    /*
        void createSub(int[][]x, int[][]y,int[][] a,int[][] b,int[][] c, int[][] d, int[][] e, int[][] f, int [][] g, int[][] h)
       int div1R = x.length/2;
       int div1C = div1R;
       int div2R = div1R;
       int div2C = div1R;
       a = helpSub(0,0,div1R,div1C,x);
      // c(a);
       b = helpSub(0,div1C,div1R,x[0].length,x);
       //c(b);
       c = helpSub(div1R,0,x.length,div1C,x);
       //c(c);
       d = helpSub(div1R,div1C,x.length,x[0].length,x);
       //c(d);
       e = helpSub(0,0,div2R,div2C,y);
       //c(e);
       f = helpSub(0,div2C,div2R,y[0].length,y);
      // c(f);
       g = helpSub(div2R,0,y.length,div2C,y);
       //c(g);
       h = helpSub(div2R,div2C,y.length,y[0].length,y);
      // c(h);


    
        */
        int[][] helpSub(int x, int y, int[][] mat)
        int[][] sub = new int[mat.length/2][mat.length/2];
        for(int i1 = 0, i2=x; i1 < (mat.length/2); i1++, i2++)
        for(int j1 = 0, j2=y; j1<(mat.length/2); j1++, j2++)
        
                sub[i1][j1] = mat[i2][j2];
                               // System.out.println(sub[i1][j1]);
        
        return sub;
    


    int[][] multiply(int[][]a,int[][]b)
        MM nu = new MM(a,b);
        return nu.product;
    

        //Adds one matrix to the next
    int[][] add(int[][]a, int[][]b)
        int [][] nu = new int[a.length][a[0].length];
        for(int i = 0; i < a.length; i++)
            for(int j = 0; j < a[i].length;j++)
                nu[i][j] = a[i][j] + b[i][j];
            
        
        return nu;
    

    //Subtracts second matrix from the first
    int[][] subtract(int[][] a, int[][] b)
        int [][] sub = new int[a.length][a.length];
        //System.out.println("made it");
        int rows = a.length;
        int columns = a[0].length;
        for(int i = 0; i < rows; i++)
            for(int j = 0; j < columns;j++)
                sub[i][j] = a[i][j] - b[i][j];
            
        
        return sub;
    

     void C(int[][] product)
        for(int i = 0; i <product.length; i++)
            for(int j = 0; j < product[i].length; j++)
                System.out.print(product[i][j]  + " ");

            
            System.out.println();