矩阵相乘问题

 
/*

将连续的若干矩阵相乘，若从左向右依次相乘，那么效率比较低。因为矩阵乘法满足结合律，所以

可以改变矩阵相乘的次序。算法将第0个到第i个矩阵从k个位置断开，利用递归找到最优的结合次序

*/

#include<iostream>

 

using namespace std;

 

const int n=6;

const int Max=1000000;

//m[i][j]表示第i个矩阵和第j个矩阵相乘的乘法总次数

//s[i][j]表示第i个矩阵和第j个矩阵相乘应该从第s[i][j]个位置断开

int M(int a[][2],int m[]
,int s[]
,int i,int j)

{

       int k,min;

       int x;

       if(i==j)

       {

              m[i][j]=0;

              s[i][j]=0;

              return 0;

       }

       else if(i<j)

       {

              min=Max;

              for(k=i;k<j;k++)

              {

                     x=M(a,m,s,i,k)+M(a,m,s,k+1,j)+a[i][0]*a[k+1][0]*a[j][1];

                     if(x<min)

                     {

                            min=x;

                            s[i][j]=k;

                     }

              }

              m[i][j]=min;

              return min;

       }

}

 

//显示结合次序

void PrintSequence(int s[]
,int i,int j)

{

       int k;

       if(i==j)

       {

              cout<<"A"<<i;

       }

       else if(i<j)

       {

              k=s[i][j];

              cout<<"(";

              PrintSequence(s,i,k);

              PrintSequence(s,k+1,j);

              cout<<")";

       }

}

 

//矩阵结构体

typedef struct

{

       int m,n;

       int **array;

}Matrix;

 

//初始化,a
[2]存放n个矩阵的行数和列数,p
存放这n个矩阵

void init(int a
[2],Matrix p
)

{

       int i,j,k;

       for(i=0;i<n;i++)

       {

              cout<<"第"<<i<<"个矩阵的行:";

              cin>>p[i].m;

              cout<<"第"<<i<<"个矩阵的列:";

              cin>>p[i].n;

              a[i][0]=p[i].m;

              a[i][1]=p[i].n;

              p[i].array=new int*[p[i].n];

              for(j=0;j<p[i].m;j++)

              {

                     p[i].array[j]=new int[p[i].n];

                     for(k=0;k<p[i].n;k++)

                     {

                            cout<<"第"<<j<<","<<k<<"个值:";

                            cin>>p[i].array[j][k];

                     }

              }

       }

}

 

void printMatrix(Matrix p
)

{

       int i,j,k;

       for(i=0;i<n;i++)

       {

               cout<<"第"<<i<<"个矩阵的行是:"<<p[i].m<<endl;

               cout<<"第"<<i<<"个矩阵的列是:"<<p[i].n<<endl;

               for(j=0;j<p[i].m;j++)

               {

                      for(k=0;k<p[i].n;k++)

                      {

                             cout<<p[i].array[j][k]<<" ";

                      }

                      cout<<endl;

               }

       }

}

 

void printMatrix(const Matrix &m)

{

       int i,j;

       cout<<"结果矩阵行是:"<<m.m<<endl;

       cout<<"结果矩阵列是:"<<m.n<<endl;

       for(i=0;i<m.m;i++)

       {

              for(j=0;j<m.n;j++)

              {

                     cout<<m.array[i][j]<<" ";

              }

              cout<<endl;

       }

}

 

//将a,b两个矩阵相乘放入矩阵c中

void Multiple(const Matrix& a,const Matrix& b,Matrix &c)

{

       if(a.n!=b.m)

       {

              cerr<<"矩阵类型不匹配!"<<endl;

              return;

       }

       int i,j,k;

       int s;

       c.m=a.m;

       c.n=b.n;

       c.array=new int*[b.n];

       for(i=0;i<a.m;i++) c.array[i]=new int[b.n];

       for(i=0;i<a.m;i++)

       {

              for(j=0;j<b.n;j++)

              {

                     s=0;

                     for(k=0;k<a.n;k++)

                     {

                            s+=a.array[i][k]*b.array[k][j];

                     }

                     c.array[i][j]=s;

              }

       }

}

 

//递归求p
矩阵乘积

void Multiple(Matrix p
,int s

,int i,int j,Matrix &result)

{

       int k;

       if(i==j)

       {

              result=p[i];

              return;

       }

       else if(i<j)

       {

              k=s[i][j];

              Matrix L,R;

              Multiple(p,s,i,k,L);

              Multiple(p,s,k+1,j,R);

              Multiple(L,R,result);

       }

}

 

void main()

{

       int a
[2];

       int m

,s

;

       Matrix p
;

       init(a,p);

    int i,j;

       cout<<M(a,m,s,0,n-1)<<endl;

       for(i=0;i<n;i++)

       {

              for(j=0;j<n;j++)

              {

                     if(i>j) m[i][j]=s[i][j]=0;

                     cout<<m[i][j]<<" ";

              }

              cout<<endl;

       }

       cout<<"-------------"<<endl;

       for(i=0;i<n;i++)

       {

              for(j=0;j<n;j++)

              {

                     cout<<s[i][j]<<" ";

              }

              cout<<endl;

       }

       PrintSequence(s,0,n-1);

       cout<<endl;

       printMatrix(p);

       Matrix result;

       Multiple(p,s,0,n-1,result);

       printMatrix(result);

}