查找一个字符串的最长子序列的长度，该字符串是另一字符串的子字符串

给定两个字符串X和Y。任务是找到字符串X的最长子序列的长度，该字符串是序列Y中的子字符串。

例子：

Input : X = "ABCD",  Y = "BACDBDCD"
Output : 3
"ACD" is longest subsequence of X which
is substring of Y.

Input : X = "A",  Y = "A"
Output : 1

方法1(蛮力)：
使用蛮力查找X的所有子序列，并为每个子序列检查它是否为Y的子串。如果它是Y的子字符串，则保持最大长度变量并将其与长度进行比较。

方法2：(动态编程)：
令n为X的长度，m为Y的长度。创建一个m + 1行和n + 1列的2D数组’dp [] []’。值dp [i] [j]是X [0….j]的子序列的最大长度，它是Y [0….i]的子串。现在，对于dp [] []的每个单元格，填充值如下：

for (i = 1 to m)
  for (j = 1 to n)
    if (x[i-1] == y[j - 1])
      dp[i][j] = dp[i-1][j-1] + 1;
    else
      dp[i][j] = dp[i][j-1];

最后，作为y子串的x最长子序列的长度为max(dp [i] [n])，其中1 <= i <= m。

下面是这种方法的实现：

C/C++

// C++ program to find maximum length of
// subsequence of a string X such it is
// substring in another string Y.
#include 
#define MAX 1000
using namespace std;
  
// Return the maximum size of substring of
// X which is substring in Y.
int maxSubsequenceSubstring(char x[], char y[],
                            int n, int m)
{
    int dp[MAX][MAX];
  
    // Initialize the dp[][] to 0.
    for (int i = 0; i <= m; i++)
        for (int j = 0; j <= n; j++)
            dp[i][j] = 0;
  
    // Calculating value for each element.
    for (int i = 1; i <= m; i++) {
        for (int j = 1; j <= n; j++) {
  
            // If alphabet of string X and Y are
            // equal make dp[i][j] = 1 + dp[i-1][j-1]
            if (x[j - 1] == y[i - 1])
                dp[i][j] = 1 + dp[i - 1][j - 1];
  
            // Else copy the previous value in the
            // row i.e dp[i-1][j-1]
            else
                dp[i][j] = dp[i][j - 1];
        }
    }
  
    // Finding the maximum length.
    int ans = 0;
    for (int i = 1; i <= m; i++)
        ans = max(ans, dp[i][n]);
  
    return ans;
}
  
// Driver Program
int main()
{
    char x[] = "ABCD";
    char y[] = "BACDBDCD";
    int n = strlen(x), m = strlen(y);
    cout << maxSubsequenceSubstring(x, y, n, m);
    return 0;
}

Java

// Java program to find maximum length of
// subsequence of a string X such it is
// substring in another string Y.
  
public class GFG 
{
    static final int MAX = 1000;
      
    // Return the maximum size of substring of
    // X which is substring in Y.
    static int maxSubsequenceSubstring(char x[], char y[],
                                int n, int m)
    {
        int dp[][] = new int[MAX][MAX];
       
        // Initialize the dp[][] to 0.
        for (int i = 0; i <= m; i++)
            for (int j = 0; j <= n; j++)
                dp[i][j] = 0;
       
        // Calculating value for each element.
        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {
       
                // If alphabet of string X and Y are
                // equal make dp[i][j] = 1 + dp[i-1][j-1]
                if (x[j - 1] == y[i - 1])
                    dp[i][j] = 1 + dp[i - 1][j - 1];
       
                // Else copy the previous value in the
                // row i.e dp[i-1][j-1]
                else
                    dp[i][j] = dp[i][j - 1];
            }
        }
       
        // Finding the maximum length.
        int ans = 0;
        for (int i = 1; i <= m; i++)
            ans = Math.max(ans, dp[i][n]);
       
        return ans;
    }
      
    // Driver Method
    public static void main(String[] args)
    {
        char x[] = "ABCD".toCharArray();
        char y[] = "BACDBDCD".toCharArray();
        int n = x.length, m = y.length;
        System.out.println(maxSubsequenceSubstring(x, y, n, m));
    }
}

Python3

# Python3 program to find maximum 
# length of subsequence of a string
# X such it is substring in another
# string Y. 
  
MAX = 1000
  
# Return the maximum size of 
# substring of X which is
# substring in Y.
def maxSubsequenceSubstring(x, y, n, m):
    dp = [[0 for i in range(MAX)]
             for i in range(MAX)]
               
    # Initialize the dp[][] to 0.
  
    # Calculating value for each element.
    for i in range(1, m + 1):
        for j in range(1, n + 1):
              
            # If alphabet of string 
            # X and Y are equal make 
            # dp[i][j] = 1 + dp[i-1][j-1]
            if(x[j - 1] == y[i - 1]):
                dp[i][j] = 1 + dp[i - 1][j - 1]
  
            # Else copy the previous value 
            # in the row i.e dp[i-1][j-1]
            else:
                dp[i][j] = dp[i][j - 1]
                  
    # Finding the maximum length
    ans = 0
    for i in range(1, m + 1):
        ans = max(ans, dp[i][n])
    return ans
  
# Driver Code 
x = "ABCD"
y = "BACDBDCD"
n = len(x)
m = len(y)
print(maxSubsequenceSubstring(x, y, n, m))
  
# This code is contributed 
# by sahilshelangia

C#

// C# program to find maximum length of
// subsequence of a string X such it is
// substring in another string Y.
using System;
  
public class GFG 
{
    static int MAX = 1000;
      
    // Return the maximum size of substring of
    // X which is substring in Y.
    static int maxSubsequenceSubstring(string x, string y,
                                            int n, int m)
    {
        int[ ,]dp = new int[MAX, MAX];
      
        // Initialize the dp[][] to 0.
        for (int i = 0; i <= m; i++)
            for (int j = 0; j <= n; j++)
                dp[i, j] = 0;
      
        // Calculating value for each element.
        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {
      
                // If alphabet of string X and Y are
                // equal make dp[i][j] = 1 + dp[i-1][j-1]
                if (x[j - 1] == y[i - 1])
                    dp[i, j] = 1 + dp[i - 1, j - 1];
      
                // Else copy the previous value in the
                // row i.e dp[i-1][j-1]
                else
                    dp[i, j] = dp[i, j - 1];
            }
        }
      
        // Finding the maximum length.
        int ans = 0;
          
        for (int i = 1; i <= m; i++)
            ans = Math.Max(ans, dp[i,n]);
      
        return ans;
    }
      
    // Driver Method
    public static void Main()
    {
        string x = "ABCD";
        string y = "BACDBDCD";
        int n = x.Length, m = y.Length;
          
        Console.WriteLine(maxSubsequenceSubstring(x,
                                            y, n, m));
    }
}
  
// This code is contributed by vt_m.

输出：

3