矩阵连乘问题

•  
• 实验目的

1、理解动态规划算法概念。

2、掌握动态规划算法的基本要素：

（1）最优子结构性质    （2）重叠子问题性质

• 实验内容

给定n个矩阵{A1, A2, …, An}，其中Ai和Ai+1是可乘的， i = 1, 2, …,n-1考察这n个矩阵的连乘积A1A2A3,…,An，使得其乘法次数最少。

实验要求：

要求计算出最优值，并构造出最优解。

实验步骤：

分析最优解的结构

建立递归关系

计算最优值

构造最优解

 

• 关键代码记录

  #include<iostream>
  using namespace std;
  int m[100][100];
  int s[100][100];
  int p[100];
  
  int MatrixChain(int* p, int n)
  {
  	for (int i = 1; i <= n; i++) { m[i][i] = 0; }//初始化单个矩阵的乘法次数都为0
  	for (int r = 2; r <= n; r++)//r为矩阵连乘的长度，即矩阵个数
  		for (int i = 1; i <= n - r + 1; i++) {//i为矩阵连乘的起点
  			int j = i + r - 1;                               //j为终点
  			m[i][j] = m[i][i] + m[i + 1][j] + p[i - 1] * p[i] * p[j];
  			s[i][j] = i;
  			for (int k = i + 1; k < j; k++) {//找出更优的乘法次数
  				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;
  				}
  			}
  		}
  	return m[1][n];
  }
  
  void print_optimal(int i, int j)
  
  {
  	if (i == j) {
  		cout << " A[" << p[i - 1] << "," << p[i] << "]";
  	}
  	else {
  		cout << " ( ";
  		print_optimal(i, s[i][j]);
  		print_optimal(s[i][j] + 1, j);
  		cout << " ) ";
  	}
  }
  int main()
  {
  	int n;
  	cout << "请输入矩阵的个数" << endl;
  	cin >> n;
  	cout << "请输入 " << n + 1 << " 个整数" << endl;
  	for (int i = 0;i <= n;i++)
  	{
  		cin >> p[i];
  	}
  	cout << "原始数据为以下矩阵" << endl;
  	for (int i = 0;i < n;i++)
  	{
  		cout << "\t" << p[i] << " * " << p[i + 1] << endl;
  	}
  	cout << "m矩阵" << endl;
  	for (int i = 1;i <= n;i++)
  	{
  		for (int j = 1;j <= n;j++)
  		{
  			m[i][j] = MatrixChain(p, n + 1);
  		}
  	}
  	for (int i = 1;i <= n;i++) {
  		for (int j = 1;j <= n;j++) {
  			cout << m[i][j] << "\t";
  		}
  		cout << endl;
  	}
  	cout << endl;
  	cout << "s矩阵" << endl;
  	for (int i = 1;i <= n;i++) {
  		for (int j = 1;j <= n;j++) {
  			cout << s[i][j] << "\t";
  		}
  		cout << endl;
  	}
  
  	cout << "最优的运算方式的乘法次数为：" << MatrixChain(p, n) << endl;
  	cout << "加括号的方式为：" << endl;
  	print_optimal(1, n);
  	cout << endl << endl;
  	system("pause");
  }
  

 测试数据：假设矩阵个数为2，五个数字为4 6 8 9 7，得出最优值为732