三个字符串的LCS（最长公共子序列）

给定 3 个长度都小于 100 的字符串，任务是在所有三个给定序列中找到最长的公共子序列。
例子：

Input : str1 = "geeks"  
        str2 = "geeksfor"  
        str3 = "geeksforgeeks"
Output : 5
Longest common subsequence is "geeks"
i.e., length = 5

Input : str1 = "abcd1e2"  
        str2 = "bc12ea"  
        str3 = "bd1ea"
Output : 3
Longest common subsequence is "b1e" 
i.e. length = 3.

这个问题简直就是LCS的扩展
设输入序列分别为长度为 m、n 和 o 的 X[0..m-1]、Y[0..n-1] 和 Z[0..o-1]。并令 L(X[0..m-1], Y[0..n-1], Z[0..o-1]) 为三个序列 X、Y 和 Z 的 LCS 长度。如下是实现：

The idea is to take a 3D array to store the 
length of common subsequence in all 3 given 
sequences i. e., L[m + 1][n + 1][o + 1]

1- If any of the string is empty then there
   is no common subsequence at all then
           L[i][j][k] = 0

2- If the characters of all sequences match
   (or X[i] == Y[j] ==Z[k]) then
        L[i][j][k] = 1 + L[i-1][j-1][k-1]

3- If the characters of both sequences do 
   not match (or X[i] != Y[j] || X[i] != Z[k] 
   || Y[j] !=Z[k]) then
        L[i][j][k] = max(L[i-1][j][k], 
                         L[i][j-1][k], 
                         L[i][j][k-1])

下面是上述想法的实现。

C++

// C++ program to find LCS of three strings
#include
using namespace std;
 
/* Returns length of LCS for X[0..m-1], Y[0..n-1]
   and Z[0..o-1] */
int lcsOf3( string X, string Y, string Z, int m,
                               int n, int o)
{
    int L[m+1][n+1][o+1];
 
    /* Following steps build L[m+1][n+1][o+1] in
       bottom up fashion. Note that L[i][j][k]
       contains length of LCS of X[0..i-1] and
       Y[0..j-1]  and Z[0.....k-1]*/
    for (int i=0; i<=m; i++)
    {
        for (int j=0; j<=n; j++)
        {
            for (int k=0; k<=o; k++)
            {
                if (i == 0 || j == 0||k==0)
                    L[i][j][k] = 0;
 
                else if (X[i-1] == Y[j-1] && X[i-1]==Z[k-1])
                    L[i][j][k] = L[i-1][j-1][k-1] + 1;
 
                else
                    L[i][j][k] = max(max(L[i-1][j][k],
                                         L[i][j-1][k]),
                                     L[i][j][k-1]);
            }
        }
    }
 
    /* L[m][n][o] contains length of LCS for
      X[0..n-1] and Y[0..m-1] and Z[0..o-1]*/
    return L[m][n][o];
}
 
/* Driver program to test above function */
int main()
{
    string X = "AGGT12";
    string Y = "12TXAYB";
    string Z = "12XBA";
 
    int m = X.length();
    int n = Y.length();
    int o = Z.length();
 
    cout << "Length of LCS is " << lcsOf3(X, Y,
                                    Z, m, n, o);
 
    return 0;
}

Java

// Java program to find LCS of three strings
public class LCS_3Strings {
      
    /* Returns length of LCS for X[0..m-1], Y[0..n-1]
       and Z[0..o-1] */
    static int lcsOf3(String X, String Y, String Z, int m,
                                   int n, int o)
    {
        int[][][] L = new int[m+1][n+1][o+1];
      
        /* Following steps build L[m+1][n+1][o+1] in
           bottom up fashion. Note that L[i][j][k]
           contains length of LCS of X[0..i-1] and
           Y[0..j-1]  and Z[0.....k-1]*/
        for (int i=0; i<=m; i++)
        {
            for (int j=0; j<=n; j++)
            {
                for (int k=0; k<=o; k++)
                {
                    if (i == 0 || j == 0||k==0)
                        L[i][j][k] = 0;
      
                    else if (X.charAt(i - 1) == Y.charAt(j - 1)
                                && X.charAt(i - 1)==Z.charAt(k - 1))
                        L[i][j][k] = L[i-1][j-1][k-1] + 1;
      
                    else
                        L[i][j][k] = Math.max(Math.max(L[i-1][j][k],
                                             L[i][j-1][k]),
                                         L[i][j][k-1]);
                }
            }
        }
      
        /* L[m][n][o] contains length of LCS for
          X[0..n-1] and Y[0..m-1] and Z[0..o-1]*/
        return L[m][n][o];
    }
      
    /* Driver program to test above function */
    public static void main(String args[])
    {
        String X = "AGGT12";
        String Y = "12TXAYB";
        String Z = "12XBA";
      
        int m = X.length();
        int n = Y.length();
        int o = Z.length();
      
        System.out.println("Length of LCS is " +
                                lcsOf3(X, Y,Z, m, n, o));
      
    }
}
// This code is contributed by Sumit Ghosh

Python3

# Python program to find
# LCS of three strings
 
# Returns length of LCS
# for X[0..m-1], Y[0..n-1]
# and Z[0..o-1]
def lcsOf3(X, Y, Z, m, n, o):
     
    L = [[[0 for i in range(o+1)] for j in range(n+1)]
         for k in range(m+1)]
 
    ''' Following steps build L[m+1][n+1][o+1] in
    bottom up fashion. Note that L[i][j][k]
    contains length of LCS of X[0..i-1] and
    Y[0..j-1] and Z[0.....k-1] '''
    for i in range(m+1):
        for j in range(n+1):
            for k in range(o+1):
                if (i == 0 or j == 0 or k == 0):
                    L[i][j][k] = 0
                     
                elif (X[i-1] == Y[j-1] and
                      X[i-1] == Z[k-1]):
                    L[i][j][k] = L[i-1][j-1][k-1] + 1
 
                else:
                    L[i][j][k] = max(max(L[i-1][j][k],
                    L[i][j-1][k]),
                                    L[i][j][k-1])
 
    # L[m][n][o] contains length of LCS for
    # X[0..n-1] and Y[0..m-1] and Z[0..o-1]
    return L[m][n][o]
 
# Driver program to test above function
 
X = 'AGGT12'
Y = '12TXAYB'
Z = '12XBA'
 
m = len(X)
n = len(Y)
o = len(Z)
 
print('Length of LCS is', lcsOf3(X, Y, Z, m, n, o))
 
# This code is contributed by Soumen Ghosh.

C#

// C# program to find
// LCS of three strings
using System;
 
class GFG
{
     
    /* Returns length of LCS
    for X[0..m-1], Y[0..n-1]
    and Z[0..o-1] */
    static int lcsOf3(String X, String Y,
                      String Z, int m,
                      int n, int o)
    {
        int [,,]L = new int[m + 1,
                            n + 1, o + 1];
     
        /* Following steps build
        L[m+1][n+1][o+1] in bottom
        up fashion. Note that
        L[i][j][k] contains length
        of LCS of X[0..i-1] and
        Y[0..j-1] and Z[0.....k-1]*/
        for (int i = 0; i <= m; i++)
        {
            for (int j = 0; j <= n; j++)
            {
                for (int k = 0; k <= o; k++)
                {
                    if (i == 0 ||
                        j == 0 || k == 0)
                        L[i, j, k] = 0;
     
                    else if (X[i - 1] == Y[j - 1] &&
                             X[i - 1] == Z[k - 1])
                        L[i, j, k] = L[i - 1,
                                       j - 1,
                                       k - 1] + 1;
     
                    else
                        L[i, j, k] = Math.Max(Math.Max(L[i - 1, j, k],
                                                       L[i, j - 1, k]),
                                                       L[i, j, k - 1]);
                }
            }
        }
     
        /* L[m][n][o] contains length
        of LCS for X[0..n-1] and
        Y[0..m-1] and Z[0..o-1]*/
        return L[m, n, o];
    }
     
    // Driver Code
    public static void Main()
    {
        string X = "AGGT12";
        string Y = "12TXAYB";
        string Z = "12XBA";
     
        int m = X.Length;
        int n = Y.Length;
        int o = Z.Length;
     
        Console.Write("Length of LCS is " +
                       lcsOf3(X, Y, Z, m, n, o));
    }
}
 
// This code is contributed
// by shiv_bhakt.

Javascript

// Javascript program to find LCS of three strings   
     
    /* Returns length of LCS for X[0..m-1], Y[0..n-1]
       and Z[0..o-1] */
    function lcsOf3(X,Y,Z,m,n,o)
    {
        let L = new Array(m + 1);
        for(let i = 0; i < m + 1; i++)
        {
            L[i] = new Array(n + 1);
            for(let j = 0; j < n + 1; j++)
            {
                L[i][j] = new Array(o + 1);
                for(let k = 0; k < o + 1; k++)
                {
                    L[i][j][k] = 0;
                }
            }
        }
         
        /* Following steps build L[m+1][n+1][o+1] in
           bottom up fashion. Note that L[i][j][k]
           contains length of LCS of X[0..i-1] and
           Y[0..j-1]  and Z[0.....k-1]*/
        for (let i = 0; i <= m; i++)
        {
            for (let j = 0; j <= n; j++)
            {
                for (let k = 0; k <= o; k++)
                {
                    if (i == 0 || j == 0 || k == 0)
                        L[i][j][k] = 0;
        
                    else if (X[i - 1] == Y[j - 1]
                                && X[i - 1] == Z[k - 1])
                        L[i][j][k] = L[i - 1][j - 1][k - 1] + 1;
        
                    else
                        L[i][j][k] = Math.max(Math.max(L[i - 1][j][k],
                                             L[i][j - 1][k]),
                                         L[i][j][k - 1]);
                }
            }
        }
        
        /* L[m][n][o] contains length of LCS for
          X[0..n-1] and Y[0..m-1] and Z[0..o-1]*/
        return L[m][n][o];
    }
     
     /* Driver program to test above function */
     let X = "AGGT12";
     let Y = "12TXAYB";
     let Z = "12XBA";
     let m = X.length;
    let n = Y.length;
    let o = Z.length;
    
    document.write("Length of LCS is " +
                            lcsOf3(X, Y,Z, m, n, o));
     
    // This code is contributed by avanitrachhadiya2155

输出：

Length of LCS is 2

另一种方法：(使用递归)

C++

// C++ program to find LCS of three strings
#include
using namespace std;
 
    string X = "AGGT12";
    string Y = "12TXAYB";
    string Z = "12XBA";
 
int dp[100][100][100];
 
/* Returns length of LCS for X[0..m-1], Y[0..n-1]
and Z[0..o-1] */
int lcsOf3(int i, int j,int k)
{
    if(i==-1||j==-1||k==-1)
        return 0;
    if(dp[i][j][k]!=-1)
        return dp[i][j][k];
     
    if(X[i]==Y[j] && Y[j]==Z[k])
        return dp[i][j][k] = 1+lcsOf3(i-1,j-1,k-1);
    else
        return dp[i][j][k] = max(max(lcsOf3(i-1,j,k),
                            lcsOf3(i,j-1,k)),lcsOf3(i,j,k-1));
}
 
// Driver code
int main()
{
    memset(dp, -1,sizeof(dp));
    int m = X.length();
    int n = Y.length();
    int o = Z.length();
 
    cout << "Length of LCS is " << lcsOf3(m-1,n-1,o-1);
// this code is contributed by Kushdeep Mittal
}

Java

// Java program to find LCS of three strings
class GFG
{
 
    static String X = "AGGT12";
    static String Y = "12TXAYB";
    static String Z = "12XBA";
 
    static int[][][] dp = new int[100][100][100];
 
    /* Returns length of LCS for X[0..m-1],
    Y[0..n-1] and Z[0..o-1] */
    static int lcsOf3(int i, int j, int k)
    {
        if (i == -1 || j == -1 || k == -1)
        {
            return 0;
        }
        if (dp[i][j][k] != -1)
        {
            return dp[i][j][k];
        }
 
        if (X.charAt(i) == Y.charAt(j) &&
            Y.charAt(j) == Z.charAt(k))
        {
            return dp[i][j][k] = 1 + lcsOf3(i - 1, j - 1, k - 1);
        } else {
            return dp[i][j][k] = Math.max(Math.max(lcsOf3(i - 1, j, k),
                                lcsOf3(i, j - 1, k)),
                                lcsOf3(i, j, k - 1));
        }
    }
 
    // Driver code
    public static void main(String[] args)
    {
 
        for (int i = 0; i < 100; i++)
        {
            for (int j = 0; j < 100; j++)
            {
                for (int k = 0; k < 100; k++)
                {
                    dp[i][j][k] = -1;
                }
            }
        }
        int m = X.length();
        int n = Y.length();
        int o = Z.length();
 
        System.out.print("Length of LCS is "
                + lcsOf3(m - 1, n - 1, o - 1));
    }
}
 
// This code has been contributed by 29AjayKumar

蟒蛇3

# Python3 program to find LCS of
# three strings
X = "AGGT12"
Y = "12TXAYB"
Z = "12XBA"
 
dp = [[[-1 for i in range(100)]
           for j in range(100)]
           for k in range(100)]
         
# Returns length of LCS for
# X[0..m-1], Y[0..n-1] and Z[0..o-1]
def lcsOf3(i, j, k) :
 
    if(i == -1 or j == -1 or k == -1) :
        return 0
         
    if(dp[i][j][k] != -1) :
        return dp[i][j][k]
     
    if(X[i] == Y[j] and Y[j] == Z[k]) :
        dp[i][j][k] = 1 + lcsOf3(i - 1,
                                 j - 1, k - 1)
        return dp[i][j][k]
         
    else :
        dp[i][j][k] = max(max(lcsOf3(i - 1, j, k),
                              lcsOf3(i, j - 1, k)),
                              lcsOf3(i, j, k - 1))
         
        return dp[i][j][k]
 
# Driver code
if __name__ == "__main__" :
    m = len(X)
    n = len(Y)
    o = len(Z)
 
    print("Length of LCS is",
           lcsOf3(m - 1, n - 1, o - 1))
     
# This code is contributed by Ryuga

C＃

// C# program to find LCS of three strings
using System;
 
class GFG
{
static string X = "AGGT12";
static string Y = "12TXAYB";
static string Z = "12XBA";
 
static int[,,] dp = new int[100, 100, 100];
 
/* Returns length of LCS for X[0..m-1],
Y[0..n-1] and Z[0..o-1] */
static int lcsOf3(int i, int j, int k)
{
    if(i == -1 || j == -1 || k == -1)
        return 0;
    if(dp[i, j, k] != -1)
        return dp[i, j, k];
     
    if(X[i] == Y[j] && Y[j] == Z[k])
        return dp[i, j, k] = 1 + lcsOf3(i - 1, j - 1, k - 1);
    else
        return dp[i, j, k] = Math.Max(Math.Max(lcsOf3(i - 1, j, k),
                                               lcsOf3(i, j - 1, k)),
                                               lcsOf3(i, j, k - 1));
}
 
// Driver code
static void Main()
{
    for(int i = 0; i < 100; i++)
        for(int j = 0; j < 100; j++)
            for(int k = 0; k < 100; k++)
                dp[i, j, k] = -1;
    int m = X.Length;
    int n = Y.Length;
    int o = Z.Length;
 
    Console.Write("Length of LCS is " +
                   lcsOf3(m - 1, n - 1, o - 1));
}
}
 
// This code is contributed by DrRoot_

Javascript

// Java program to find LCS of three strings
 
let X = "AGGT12";
let Y = "12TXAYB";
let Z = "12XBA";
 
let dp = new Array(100);
for(let i=0;i<100;i++)
{
    dp[i]=new Array(100);
    for(let j=0;j<100;j++)
    {
        dp[i][j]=new Array(100);
        for(let k=0;k<100;k++)
        {
            dp[i][j][k]=-1;
        }
    }
}
 
/* Returns length of LCS for X[0..m-1],
    Y[0..n-1] and Z[0..o-1] */
function lcsOf3(i,j,k)
{
     if (i == -1 || j == -1 || k == -1)
        {
            return 0;
        }
        if (dp[i][j][k] != -1)
        {
            return dp[i][j][k];
        }
  
        if (X[i] == Y[j] &&
            Y[j] == Z[k])
        {
            return dp[i][j][k] = 1 + lcsOf3(i - 1, j - 1, k - 1);
        } else {
            return dp[i][j][k] = Math.max(Math.max(lcsOf3(i - 1, j, k),
                                lcsOf3(i, j - 1, k)),
                                lcsOf3(i, j, k - 1));
        }
}
 
// Driver code
 
        let m = X.length;
        let n = Y.length;
        let o = Z.length;
  
        document.write("Length of LCS is "
                + lcsOf3(m - 1, n - 1, o - 1));
 
 
// This code is contributed by rag2127

输出：

Length of LCS is 2

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