Longest Common Substring | DP-29

Given two strings ‘X’ and ‘Y’, find the length of the longest common substring. 

Examples :

Input : X = “GeeksforGeeks”, y = “GeeksQuiz” 
Output : 5 
Explanation:
The longest common substring is “Geeks” and is of length 5.

Input : X = “abcdxyz”, y = “xyzabcd” 
Output : 4 
Explanation:
The longest common substring is “abcd” and is of length 4.

Input : X = “zxabcdezy”, y = “yzabcdezx” 
Output : 6 
Explanation:
The longest common substring is “abcdez” and is of length 6.

Approach:
Let m and n be the lengths of the first and second strings respectively.

A simple solution is to one by one consider all substrings of the first string and for every substring check if it is a substring in the second string. Keep track of the maximum length substring. There will be O(m^2) substrings and we can find whether a string is substring on another string in O(n) time (See this). So overall time complexity of this method would be O(n * m²)

Dynamic Programming can be used to find the longest common substring in O(m*n) time. The idea is to find the length of the longest common suffix for all substrings of both strings and store these lengths in a table. 

The longest common suffix has following optimal substructure property. 

If last characters match, then we reduce both lengths by 1 

• LCSuff(X, Y, m, n) = LCSuff(X, Y, m-1, n-1) + 1 if X[m-1] = Y[n-1] 

If last characters do not match, then result is 0, i.e., 

• LCSuff(X, Y, m, n) = 0 if (X[m-1] != Y[n-1])

Now we consider suffixes of different substrings ending at different indexes. 
The maximum length Longest Common Suffix is the longest common substring. 
LCSubStr(X, Y, m, n) = Max(LCSuff(X, Y, i, j)) where 1 <= i <= m and 1 <= j <= n 

Following is the iterative implementation of the above solution.  

C++

/* Dynamic Programming solution to
   find length of the
   longest common substring */
#include <iostream>
#include <string.h>
using namespace std;

/* Returns length of longest
   common substring of X[0..m-1]
   and Y[0..n-1] */
int LCSubStr(char* X, char* Y, int m, int n)
{
    // Create a table to store
    // lengths of longest
    // common suffixes of substrings.
    // Note that LCSuff[i][j] contains
    // length of longest common suffix
    // of X[0..i-1] and Y[0..j-1].

    int LCSuff[m + 1][n + 1];
    int result = 0; // To store length of the
                    // longest common substring

    /* Following steps build LCSuff[m+1][n+1] in
        bottom up fashion. */
    for (int i = 0; i <= m; i++)
    {
        for (int j = 0; j <= n; j++)
        {
            // The first row and first column
            // entries have no logical meaning,
            // they are used only for simplicity
            // of program
            if (i == 0 || j == 0)
                LCSuff[i][j] = 0;

            else if (X[i - 1] == Y[j - 1]) {
                LCSuff[i][j] = LCSuff[i - 1][j - 1] + 1;
                result = max(result, LCSuff[i][j]);
            }
            else
                LCSuff[i][j] = 0;
        }
    }
    return result;
}

// Driver code
int main()
{
    char X[] = "OldSite:GeeksforGeeks.org";
    char Y[] = "NewSite:GeeksQuiz.com";

    int m = strlen(X);
    int n = strlen(Y);

    cout << "Length of Longest Common Substring is "
         << LCSubStr(X, Y, m, n);
    return 0;
}

Java

//  Java implementation of
// finding length of longest
// Common substring using
// Dynamic Programming
import java.io.*;

class GFG {
    /*
       Returns length of longest common substring
       of X[0..m-1] and Y[0..n-1]
    */
    static int LCSubStr(char X[], char Y[], int m, int n)
    {
        // Create a table to store
        // lengths of longest common
        // suffixes of substrings.
        // Note that LCSuff[i][j]
        // contains length of longest
        // common suffix of
        // X[0..i-1] and Y[0..j-1].
        // The first row and first
        // column entries have no
        // logical meaning, they are
        // used only for simplicity of program
        int LCStuff[][] = new int[m + 1][n + 1];

        // To store length of the longest
        // common substring
        int result = 0;

        // Following steps build
        // LCSuff[m+1][n+1] in bottom up fashion
        for (int i = 0; i <= m; i++) {
            for (int j = 0; j <= n; j++) {
                if (i == 0 || j == 0)
                    LCStuff[i][j] = 0;
                else if (X[i - 1] == Y[j - 1]) {
                    LCStuff[i][j]
                        = LCStuff[i - 1][j - 1] + 1;
                    result = Integer.max(result,
                                         LCStuff[i][j]);
                }
                else
                    LCStuff[i][j] = 0;
            }
        }
        return result;
    }

    // Driver Code
    public static void main(String[] args)
    {
        String X = "OldSite:GeeksforGeeks.org";
        String Y = "NewSite:GeeksQuiz.com";

        int m = X.length();
        int n = Y.length();

        System.out.println(
            "Length of Longest Common Substring is "
            + LCSubStr(X.toCharArray(), Y.toCharArray(), m,
                       n));
    }
}

// This code is contributed by Sumit Ghosh

Python3

# Python3 implementation of Finding
# Length of Longest Common Substring

# Returns length of longest common
# substring of X[0..m-1] and Y[0..n-1]


def LCSubStr(X, Y, m, n):

    # Create a table to store lengths of
    # longest common suffixes of substrings.
    # Note that LCSuff[i][j] contains the
    # length of longest common suffix of
    # X[0...i-1] and Y[0...j-1]. The first
    # row and first column entries have no
    # logical meaning, they are used only
    # for simplicity of the program.

    # LCSuff is the table with zero
    # value initially in each cell
    LCSuff = [[0 for k in range(n+1)] for l in range(m+1)]

    # To store the length of
    # longest common substring
    result = 0

    # Following steps to build
    # LCSuff[m+1][n+1] in bottom up fashion
    for i in range(m + 1):
        for j in range(n + 1):
            if (i == 0 or j == 0):
                LCSuff[i][j] = 0
            elif (X[i-1] == Y[j-1]):
                LCSuff[i][j] = LCSuff[i-1][j-1] + 1
                result = max(result, LCSuff[i][j])
            else:
                LCSuff[i][j] = 0
    return result


# Driver Code
X = 'OldSite:GeeksforGeeks.org'
Y = 'NewSite:GeeksQuiz.com'

m = len(X)
n = len(Y)

print('Length of Longest Common Substring is',
      LCSubStr(X, Y, m, n))

# This code is contributed by Soumen Ghosh

C#

// C# implementation of finding length of longest
// Common substring using Dynamic Programming
using System;

class GFG {

    // Returns length of longest common
    // substring of X[0..m-1] and Y[0..n-1]
    static int LCSubStr(string X, string Y, int m, int n)
    {

        // Create a table to store lengths of
        // longest common suffixes of substrings.
        // Note that LCSuff[i][j] contains length
        // of longest common suffix of X[0..i-1]
        // and Y[0..j-1]. The first row and first
        // column entries have no logical meaning,
        // they are used only for simplicity of
        // program
        int[, ] LCStuff = new int[m + 1, n + 1];

        // To store length of the longest common
        // substring
        int result = 0;

        // Following steps build LCSuff[m+1][n+1]
        // in bottom up fashion
        for (int i = 0; i <= m; i++)
        {
            for (int j = 0; j <= n; j++)
            {
                if (i == 0 || j == 0)
                    LCStuff[i, j] = 0;
                else if (X[i - 1] == Y[j - 1])
                {
                    LCStuff[i, j]
                        = LCStuff[i - 1, j - 1] + 1;

                    result
                        = Math.Max(result, LCStuff[i, j]);
                }
                else
                    LCStuff[i, j] = 0;
            }
        }

        return result;
    }

    // Driver Code
    public static void Main()
    {
        String X = "OldSite:GeeksforGeeks.org";
        String Y = "NewSite:GeeksQuiz.com";

        int m = X.Length;
        int n = Y.Length;

        Console.Write("Length of Longest Common"
                      + " Substring is "
                      + LCSubStr(X, Y, m, n));
    }
}

// This code is contributed by Sam007.

PHP

<?php
// Dynamic Programming solution to find
// length of the longest common substring

// Returns length of longest common
// substring of X[0..m-1] and Y[0..n-1]
function LCSubStr($X, $Y, $m, $n)
{
    // Create a table to store lengths of
    // longest common suffixes of substrings.
    // Notethat LCSuff[i][j] contains length
    // of longest common suffix of X[0..i-1]
    // and Y[0..j-1]. The first row and
    // first column entries have no logical
    // meaning, they are used only for
    // simplicity of program
    $LCSuff = array_fill(0, $m + 1,
              array_fill(0, $n + 1, NULL));
    $result = 0; // To store length of the
                 // longest common substring

    // Following steps build LCSuff[m+1][n+1]
    // in bottom up fashion.
    for ($i = 0; $i <= $m; $i++)
    {
        for ($j = 0; $j <= $n; $j++)
        {
            if ($i == 0 || $j == 0)
                $LCSuff[$i][$j] = 0;

            else if ($X[$i - 1] == $Y[$j - 1])
            {
                $LCSuff[$i][$j] = $LCSuff[$i - 1][$j - 1] + 1;
                $result = max($result,
                              $LCSuff[$i][$j]);
            }
            else $LCSuff[$i][$j] = 0;
        }
    }
    return $result;
}

// Driver Code
$X = "OldSite:GeeksforGeeks.org";
$Y = "NewSite:GeeksQuiz.com";

$m = strlen($X);
$n = strlen($Y);

echo "Length of Longest Common Substring is " .
                      LCSubStr($X, $Y, $m, $n);

// This code is contributed by ita_c
?>

Javascript

<script>

// JavaScript implementation of
// finding length of longest
// Common substring using
// Dynamic Programming

    /*
     Returns length of longest common
     substring of X[0..m-1] and Y[0..n-1]
     */
    function LCSubStr( X,  Y , m , n) {
        // Create a table to store
        // lengths of longest common
        // suffixes of substrings.
        // Note that LCSuff[i][j]
        // contains length of longest
        // common suffix of
        // X[0..i-1] and Y[0..j-1].
        // The first row and first
        // column entries have no
        // logical meaning, they are
        // used only for simplicity of program

        var LCStuff =
        Array(m + 1).fill().map(()=>Array(n + 1).fill(0));

        // To store length of the longest
        // common substring
        var result = 0;

        // Following steps build
        // LCSuff[m+1][n+1] in bottom up fashion
        for (i = 0; i <= m; i++) {
            for (j = 0; j <= n; j++) {
                if (i == 0 || j == 0)
                    LCStuff[i][j] = 0;
                else if (X[i - 1] == Y[j - 1]) {
                    LCStuff[i][j] = LCStuff[i - 1][j - 1] + 1;
                    result = Math.max(result, LCStuff[i][j]);
                } else
                    LCStuff[i][j] = 0;
            }
        }
        return result;
    }

    // Driver Code

        var X = "OldSite:GeeksforGeeks.org";
        var Y = "NewSite:GeeksQuiz.com";

        var m = X.length;
        var n = Y.length;

        document.write("Length of Longest Common Substring is " +
        LCSubStr(X, Y, m, n));

// This code contributed by Rajput-Ji

</script>

Output

Length of Longest Common Substring is 10

Time Complexity: O(m*n) 
Auxiliary Space: O(m*n), since m*n extra space has been taken.

Another Approach: (Space optimized approach).
In the above approach, we are only using the last row of the 2-D array only, hence we can optimize the space by using 
a 2-D array of dimension 2*(min(n,m)).

Below is the implementation of the above approach:

C++

// C++ implementation of the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to find the length of the
// longest LCS
int LCSubStr(string s, string t, int n, int m)
{

    // Create DP table
    int dp[2][m + 1];
    int res = 0;

      dp[0][0] = 0;
      dp[1][0] = 0;

    for (int i = 1; i <= n; i++) {
        for (int j = 1; j <= m; j++) {
            if (s[i - 1] == t[j - 1]) {
                dp[i % 2][j] = dp[(i - 1) % 2][j - 1] + 1;
                if (dp[i % 2][j] > res)
                    res = dp[i % 2][j];
            }
            else
                dp[i % 2][j] = 0;
        }
    }
    return res;
}

// Driver Code
int main()
{
    string X = "OldSite:GeeksforGeeks.org";
    string Y = "NewSite:GeeksQuiz.com";

    int m = X.length();
    int n = Y.length();

    cout << LCSubStr(X, Y, m, n);
    return 0;
    cout << "GFG!";
    return 0;
}

// This code is contributed by rajsanghavi9.

Java

// Java implementation of the above approach
import java.io.*;
class GFG
{

    // Function to find the length of the
    // longest LCS
    static int LCSubStr(String s,String t,
                        int n,int m)
    {

        // Create DP table
        int dp[][]=new int[2][m+1];
        int res=0;

        for(int i=1;i<=n;i++)
        {
            for(int j=1;j<=m;j++)
            {
                if(s.charAt(i-1)==t.charAt(j-1))
                {
                    dp[i%2][j]=dp[(i-1)%2][j-1]+1;
                    if(dp[i%2][j]>res)
                        res=dp[i%2][j];
                }
                else dp[i%2][j]=0;
            }
        }
        return res;
    }

    // Driver Code
    public static void main (String[] args)
    {
        String X="OldSite:GeeksforGeeks.org";
        String Y="NewSite:GeeksQuiz.com";

        int m=X.length();
        int n=Y.length();

        // Function call
        System.out.println(LCSubStr(X,Y,m,n));

    }
}

Python3

# Python implementation of the above approach

# Function to find the length of the
# longest LCS
def LCSubStr(s, t, n, m):

    # Create DP table
    dp = [[0 for i in range(m + 1)] for j in range(2)]
    res = 0

    for i in range(1,n + 1):
        for j in range(1,m + 1):
            if(s[i - 1] == t[j - 1]):
                dp[i % 2][j] = dp[(i - 1) % 2][j - 1] + 1
                if(dp[i % 2][j] > res):
                    res = dp[i % 2][j]
            else:
                dp[i % 2][j] = 0
    return res

# Driver Code
X = "OldSite:GeeksforGeeks.org"
Y = "NewSite:GeeksQuiz.com"
m = len(X)
n = len(Y)

# Function call
print(LCSubStr(X,Y,m,n))

# This code is contributed by avanitrachhadiya2155

C#

// C# implementation of the above approach
using System;
public class GFG
{

  // Function to find the length of the
  // longest LCS
  static int LCSubStr(string s,string t,
                      int n,int m)
  {

    // Create DP table
    int[,] dp = new int[2, m + 1];
    int res = 0;

    for(int i = 1; i <= n; i++)
    {
      for(int j = 1; j <= m; j++)
      {
        if(s[i - 1] == t[j - 1])
        {
          dp[i % 2, j] = dp[(i - 1) % 2, j - 1] + 1;
          if(dp[i % 2, j] > res)
            res = dp[i % 2, j];
        }
        else dp[i % 2, j] = 0;
      }
    }
    return res;
  }

  // Driver Code
  static public void Main (){
    string X = "OldSite:GeeksforGeeks.org";
    string Y = "NewSite:GeeksQuiz.com";

    int m = X.Length;
    int n = Y.Length;

    // Function call
    Console.WriteLine(LCSubStr(X,Y,m,n));
  }
}

// This code is contributed by rag2127

Javascript

<script>
// JavaScript implementation of the above approach

    // Function to find the length of the
    // longest LCS
    function LCSubStr(s, t, n, m)
    {

        // Create DP table
        var dp = Array(2).fill().map(()=>Array(m+ 1).fill(0));
        var res = 0;

        for(var i = 1; i <= n; i++)
        {
            for(var j = 1; j <= m; j++)
            {
                if(s.charAt(i - 1) == t.charAt(j - 1))
                {
                    dp[i % 2][j] = dp[(i - 1) % 2][j - 1] + 1;
                    if(dp[i % 2][j] > res)
                        res = dp[i % 2][j];
                }
                else dp[i % 2][j] = 0;
            }
        }
        return res;
    }

    // Driver Code
        var X = "OldSite:GeeksforGeeks.org";
        var Y = "NewSite:GeeksQuiz.com";

        var m = X.length;
        var n = Y.length;

        // Function call
        document.write(LCSubStr(X, Y, m, n));

// This code is contributed by shivanisinghss2110
</script>

Output

10

Time Complexity: O(n*m)
Auxiliary Space: O(min(m,n))

Another Approach: (Using recursion) 
Here is the recursive solution of the above approach. 

C++

// C++ program using to find length of the
// longest common substring  recursion
#include <iostream>

using namespace std;

string X, Y;

// Returns length of function f
// or longest common substring
// of X[0..m-1] and Y[0..n-1]
int lcs(int i, int j, int count)
{

    if (i == 0 || j == 0)
        return count;

    if (X[i - 1] == Y[j - 1]) {
        count = lcs(i - 1, j - 1, count + 1);
    }
    count = max(count,
                max(lcs(i, j - 1, 0),
                    lcs(i - 1, j, 0)));
    return count;
}

// Driver code
int main()
{
    int n, m;

    X = "abcdxyz";
    Y = "xyzabcd";

    n = X.size();
    m = Y.size();

    cout << lcs(n, m, 0);

    return 0;
}

Java

// Java program using to find length of the
// longest common substring recursion
import java.io.*;
class GFG {

    static String X, Y;
    // Returns length of function
    // for longest common
    // substring of X[0..m-1] and Y[0..n-1]
    static int lcs(int i, int j, int count)
    {

        if (i == 0 || j == 0)
        {
            return count;
        }

        if (X.charAt(i - 1)
            == Y.charAt(j - 1))
        {
            count = lcs(i - 1, j - 1, count + 1);
        }
        count = Math.max(count,
                         Math.max(lcs(i, j - 1, 0),
                                  lcs(i - 1, j, 0)));
        return count;
    }

    // Driver code
    public static void main(String[] args)
    {
        int n, m;
        X = "abcdxyz";
        Y = "xyzabcd";

        n = X.length();
        m = Y.length();

        System.out.println(lcs(n, m, 0));
    }
}
// This code is contributed by Rajput-JI

Python3

# Python3 program using to find length of
# the longest common substring recursion

# Returns length of function for longest
# common substring of X[0..m-1] and Y[0..n-1]


def lcs(i, j, count):

    if (i == 0 or j == 0):
        return count

    if (X[i - 1] == Y[j - 1]):
        count = lcs(i - 1, j - 1, count + 1)

    count = max(count, max(lcs(i, j - 1, 0),
                           lcs(i - 1, j, 0)))

    return count


# Driver code
if __name__ == "__main__":

    X = "abcdxyz"
    Y = "xyzabcd"

    n = len(X)
    m = len(Y)

    print(lcs(n, m, 0))

# This code is contributed by Ryuga

C#

// C# program using to find length
// of the longest common substring
// recursion
using System;

class GFG {
    static String X, Y;

    // Returns length of function for
    // longest common substring of
    // X[0..m-1] and Y[0..n-1]
    static int lcs(int i, int j, int count)
    {

        if (i == 0 || j == 0) {
            return count;
        }

        if (X[i - 1] == Y[j - 1]) {
            count = lcs(i - 1, j - 1, count + 1);
        }
        count = Math.Max(count, Math.Max(lcs(i, j - 1, 0),
                                         lcs(i - 1, j, 0)));
        return count;
    }

    // Driver code
    public static void Main()
    {
        int n, m;
        X = "abcdxyz";
        Y = "xyzabcd";

        n = X.Length;
        m = Y.Length;

        Console.Write(lcs(n, m, 0));
    }
}

// This code is contributed by Rajput-JI

PHP

<?php
// PHP program using to find length of the
// longest common substring recursion

// Returns length of function for
// longest common substring of
// X[0..m-1] and Y[0..n-1]
function lcs($i, $j, $count, &$X, &$Y)
{
    if ($i == 0 || $j == 0)
        return $count;

    if ($X[$i - 1] == $Y[$j - 1])
    {
        $count = lcs($i - 1, $j - 1,
                     $count + 1, $X, $Y);
    }
        $count = max($count, lcs($i, $j - 1, 0, $X, $Y),
                             lcs($i - 1, $j, 0, $X, $Y));
    return $count;
}

// Driver code
$X = "abcdxyz";
$Y = "xyzabcd";

$n = strlen($X);
$m = strlen($Y);

echo lcs($n, $m, 0, $X, $Y);

// This code is contributed
// by rathbhupendra
?>

Javascript

<script>
    // Javascript program using to find length of the
    // longest common substring  recursion
    let X, Y;

    // Returns length of function f
    // or longest common substring
    // of X[0..m-1] and Y[0..n-1]
    function lcs(i, j, count)
    {

        if (i == 0 || j == 0)
            return count;

        if (X[i - 1] == Y[j - 1]) {
            count = lcs(i - 1, j - 1, count + 1);
        }
        count = Math.max(count,
                    Math.max(lcs(i, j - 1, 0),
                        lcs(i - 1, j, 0)));
        return count;
    }

    let n, m;

    X = "abcdxyz";
    Y = "xyzabcd";

    n = X.length;
    m = Y.length;

    document.write(lcs(n, m, 0));

    // This code is contributed by divyeshrabadiya07.
</script>

Output

4

Time complexity: O(2^max(m,n)) as the function is doing two recursive calls – lcs(i, j-1, 0) and lcs(i-1, j, 0) when characters at X[i-1] != Y[j-1]. So it will give a worst case time complexity as 2^N, where N = max(m, n), m and n is the length of X and Y string.
Auxiliary Space: O(1): as the function call is not using any extra space (function is just using a recursive call stack which we generally doesn’t consider in auxiliary space).

Maximum Space Optimization:Ad