矩阵连乘（DP动态规划）

版权声明：@ly https://blog.csdn.net/lytwy123/article/details/83062758

#include<iostream>
#include<cstdio>
using namespace std;
int p[101];			//存放各个矩阵的行列值，比如，矩阵A[i]的行列值为p(i-1)*p(i) 
int m[101][101];	//存储存放最少数乘次数 
int s[101][101];    //存储加括号的位置  

void MatrixChain(int *p,int n,int m[][101],int s[][101]) //n表示有多少个矩阵 
{
	for(int i = 1;i<=n;i++)     
		m[i][i] = 0;
		for(int r = 2;r<=n;r++){
			for(int i = 1;i<=n-r+1;i++){
				int j = i + r -1;
				m[i][j] = m[i+1][j] + p[i-1]*p[i]*p[j];
				s[i][j] = i;
				for(int k = i+1;k<j;k++)
				{
					int t = m[i][k] + m[k+1][j]+p[i-1]*p[k]*p[j];
					if(t<m[i][j])
					{
						m[i][j] = t;
						s[i][j] = k;
					}
				}
			} 
		} 
}

void traceback(int i,int j,int s[][101])//递归矩阵连乘次序 
{
	if(i == j)
	cout<<"A"<<i;
	else if(i == j-1)
	cout<<"(A"<<i<<"A"<<j<<")";
	else
	{
		cout<<"(";
		traceback(i,s[i][j],s);
		traceback(s[i][j]+1,j,s);
		cout<<")";
	}
}

int main()
{
	int n;
	cin>>n;
	cout<<"please input p[]:"<<endl;
	for(int i = 0;i<=n;i++)
	cin>>p[i];
	MatrixChain(p,n,m,s);
	cout<<"输出矩阵的数乘次数:"<<endl; 
	for(int i = 1;i<=n;i++)
	{
		for(int j = 1;j<=n;j++){
			cout<<m[i][j]<<",";
			if(j == n)
			cout<<endl;
		}
	}
	cout<<"输出矩阵的加括号位置："<<endl;
	for(int i = 1;i<=n;i++)
	{
		for(int j = 1;j<=n;j++){
			cout<<s[i][j]<<",";
			if(j == n)
			cout<<endl;
		}
	}
	
	cout<<"输出矩阵最小连乘的值："<<endl;
	cout<<m[1][n];
	cout<<endl;
	cout<<"矩阵最小连乘所加括号位置："<<endl;
	cout<<s[1][n];
	cout<<"矩阵最小连乘表达式："<<endl;
	traceback(1,n,s);
	return 0;
}