动态规划二：最长公共子序列（LCS）

　　1.两个子序列：X={x₁,x₂....x_m}，Y={y_1,y_2....y_n}，设Z={z₁,z₂...z_k}。

　　2.最优子结构：

　　1）如果x_m=y_n ，则z_k=x_m=y_n且Z_k-1是X_m-1和Y_n-1的一个LCS。

　　2）如果x_m!=y_n ，则z_k!=x_m包含Z是X_m-1和Y的一个LCS。

　　3）如果x_m!=y_n ，则z_k!=y_n包含Z是X和Y_n-1的一个LCS。

　　3.则由最优子结构可得递归式：

　　aaarticlea/png;base64,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" alt="" />

　　4.实现：

 #include<string>
 #include<iostream>
 using namespace std;

 int arr[][] = {};
 void LCS(string str1, string str2){
     int len1 = str1.length();
     int len2 = str2.length();
     //边界
     for (int i = ; i <= len1; i++)
         arr[i][] = ;
     for (int j = ; j <= len2; j++)
         arr[][j] = ;
     //检验
     for (int i = ; i <= len1; i++)
     {
         for (int j = ; j <= len2; j++)
         {
             if (str1[i - ] == str2[j - ]) //记录
             {
                 arr[i][j] = arr[i - ][j - ] + ;
             }
             else if (arr[i][j - ] >= arr[i - ][j])
             {
                 arr[i][j] = arr[i][j - ];
             }
             else
             {
                 arr[i][j] = arr[i - ][j];
             }
         }
     }
 }

 void printLCS(string str1, string str2,int len1, int len2){
     cout << "最长公共子序列长为：" << arr[len1][len2] << endl;
     //倒序输出
     while (len1&&len2){
         if (str1[len1-]==str2[len2-])
         {
             cout << str1[len1 - ];
             len1--;
             len2--;
         }
         else if (arr[len1][len2-]>=arr[len1-][len2])
         {
             len2--;
         }
         else{
             len1--;
         }
     }
     cout << endl;
 }

 int main()
 {
     string str1 = "abcdsnnnn";
     string str2 = "acbdanm";
     LCS(str1, str2);
     printLCS(str1, str2,str1.length(), str2.length());
     return ;
 }