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Printing Shortest Common Supersequence

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Given two strings X and Y, print the shortest string that has both X and Y as subsequences. If multiple shortest super-sequence exists, print any one of them.
Examples: 
 

Input: X = "AGGTAB",  Y = "GXTXAYB"
Output: "AGXGTXAYB" OR "AGGXTXAYB"
OR Any string that represents shortest
supersequence of X and Y

Input: X = "HELLO", Y = "GEEK"
Output: "GEHEKLLO" OR "GHEEKLLO"
OR Any string that represents shortest
supersequence of X and Y


 


We have discussed how to print length of shortest possible super-sequence for two given strings here. In this post, we print the shortest super-sequence.
We have already discussed below algorithm to find length of shortest super-sequence in previous post-
 

Let X[0..m-1] and Y[0..n-1] be two strings and m and be respective 
lengths.

if (m == 0) return n;
if (n == 0) return m;

// If last characters are same, then add 1 to result and
// recur for X[]
if (X[m-1] == Y[n-1])
return 1 + SCS(X, Y, m-1, n-1);

// Else find shortest of following two
// a) Remove last character from X and recur
// b) Remove last character from Y and recur
else return 1 + min( SCS(X, Y, m-1, n), SCS(X, Y, m, n-1) );


The following table shows steps followed by the above algorithm if we solve it in bottom-up manner using Dynamic Programming for strings X = “AGGTAB” and Y = “GXTXAYB”
 

Shortest Supersequence Problem DP table


Using the DP solution matrix, we can easily print shortest super-sequence of two strings by following below steps –
 

We start from the bottom-right most cell of the matrix and 
push characters in output string based on below rules-

1. If the characters corresponding to current cell (i, j)
in X and Y are same, then the character is part of shortest
supersequence. We append it in output string and move
diagonally to next cell (i.e. (i - 1, j - 1)).

2. If the characters corresponding to current cell (i, j)
in X and Y are different, we have two choices -

If matrix[i - 1][j] > matrix[i][j - 1],
we add character corresponding to current
cell (i, j) in string Y in output string
and move to the left cell i.e. (i, j - 1)
else
we add character corresponding to current
cell (i, j) in string X in output string
and move to the top cell i.e. (i - 1, j)

3. If string Y reaches its end i.e. j = 0, we add remaining
characters of string X in the output string
else if string X reaches its end i.e. i = 0, we add
remaining characters of string Y in the output string.


Below is the implementation of above idea – 
 

C++14

/* A dynamic programming based C++ program print
   shortest supersequence of two strings */
#include <bits/stdc++.h>
using namespace std;
 
// returns shortest supersequence of X and Y
string printShortestSuperSeq(string X, string Y)
{
    int m = X.length();
    int n = Y.length();
 
    // dp[i][j] contains length of shortest supersequence
    // for X[0..i-1] and Y[0..j-1]
    int dp[m + 1][n + 1];
 
    // Fill table in bottom up manner
    for (int i = 0; i <= m; i++)
    {
        for (int j = 0; j <= n; j++)
        {
            // Below steps follow recurrence relation
            if(i == 0)
                dp[i][j] = j;
            else if(j == 0)
                dp[i][j] = i;
            else if(X[i - 1] == Y[j - 1])
                dp[i][j] = 1 + dp[i - 1][j - 1];
            else
                dp[i][j] = 1 + min(dp[i - 1][j], dp[i][j - 1]);
        }
    }
 
    // Following code is used to print shortest supersequence
 
    // dp[m][n] stores the length of the shortest supersequence
    // of X and Y
     
 
    // string to store the shortest supersequence
    string str;
 
    // Start from the bottom right corner and one by one
    // push characters in output string
    int i = m, j = n;
    while (i > 0 && j > 0)
    {
        // If current character in X and Y are same, then
        // current character is part of shortest supersequence
        if (X[i - 1] == Y[j - 1])
        {
            // Put current character in result
            str.push_back(X[i - 1]);
 
            // reduce values of i, j and index
            i--, j--;
        }
 
        // If current character in X and Y are different
        else if (dp[i - 1][j] > dp[i][j - 1])
        {
            // Put current character of Y in result
            str.push_back(Y[j - 1]);
 
            // reduce values of j and index
            j--;
        }
        else
        {
            // Put current character of X in result
            str.push_back(X[i - 1]);
 
            // reduce values of i and index
            i--;
        }
    }
 
    // If Y reaches its end, put remaining characters
    // of X in the result string
    while (i > 0)
    {
        str.push_back(X[i - 1]);
        i--;
    }
 
    // If X reaches its end, put remaining characters
    // of Y in the result string
    while (j > 0)
    {
        str.push_back(Y[j - 1]);
        j--;
    }
 
    // reverse the string and return it
    reverse(str.begin(), str.end());
    return str;
}
 
// Driver program to test above function
int main()
{
    string X = "AGGTAB";
    string Y = "GXTXAYB";
 
    cout << printShortestSuperSeq(X, Y);
 
    return 0;
}

                    

Java

/* A dynamic programming based Java program print
shortest supersequence of two strings */
class GFG {
 
    // returns shortest supersequence of X and Y
    static String printShortestSuperSeq(String X, String Y)
    {
        int m = X.length();
        int n = Y.length();
 
        // dp[i][j] contains length of
        // shortest supersequence
        // for X[0..i-1] and Y[0..j-1]
        int dp[][] = new int[m + 1][n + 1];
 
        // Fill table in bottom up manner
        for (int i = 0; i <= m; i++)
        {
            for (int j = 0; j <= n; j++)
            {
                 
                // Below steps follow recurrence relation
                if (i == 0)
                {
                    dp[i][j] = j;
                }
                else if (j == 0)
                {
                    dp[i][j] = i;
                }
                else if (X.charAt(i - 1) == Y.charAt(j - 1))
                {
                    dp[i][j] = 1 + dp[i - 1][j - 1];
                }
                else
                {
                    dp[i][j] = 1 + Math.min(dp[i - 1][j], dp[i][j - 1]);
                }
            }
        }
 
        // Following code is used to print
        // shortest supersequence dp[m][n] s
        // tores the length of the shortest
        // supersequence of X and Y
 
        // string to store the shortest supersequence
        String str = "";
 
        // Start from the bottom right corner and one by one
        // push characters in output string
        int i = m, j = n;
        while (i > 0 && j > 0)
         
        {
            // If current character in X and Y are same, then
            // current character is part of shortest supersequence
            if (X.charAt(i - 1) == Y.charAt(j - 1))
             
            {
                // Put current character in result
                str += (X.charAt(i - 1));
 
                // reduce values of i, j and index
                i--;
                j--;
            }
             
            // If current character in X and Y are different
            else if (dp[i - 1][j] > dp[i][j - 1])
            {
                 
                // Put current character of Y in result
                str += (Y.charAt(j - 1));
 
                // reduce values of j and index
                j--;
            }
            else
            {
                 
                // Put current character of X in result
                str += (X.charAt(i - 1));
 
                // reduce values of i and index
                i--;
            }
        }
 
        // If Y reaches its end, put remaining characters
        // of X in the result string
        while (i > 0)
        {
            str += (X.charAt(i - 1));
            i--;
        }
 
        // If X reaches its end, put remaining characters
        // of Y in the result string
        while (j > 0)
        {
            str += (Y.charAt(j - 1));
            j--;
        }
 
        // reverse the string and return it
        str = reverse(str);
        return str;
    }
 
    static String reverse(String input)
    {
        char[] temparray = input.toCharArray();
        int left, right = 0;
        right = temparray.length - 1;
 
        for (left = 0; left < right; left++, right--)
        {
            // Swap values of left and right
            char temp = temparray[left];
            temparray[left] = temparray[right];
            temparray[right] = temp;
        }
        return String.valueOf(temparray);
    }
     
    // Driver code
    public static void main(String[] args)
    {
        String X = "AGGTAB";
        String Y = "GXTXAYB";
        System.out.println(printShortestSuperSeq(X, Y));
    }
}
 
// This code is contributed by 29AjayKumar

                    

Python3

# A dynamic programming based Python3 program print
# shortest supersequence of two strings
 
# returns shortest supersequence of X and Y
 
 
def printShortestSuperSeq(m, n, x, y):
 
    # dp[i][j] contains length of shortest
    # supersequence for X[0..i-1] and Y[0..j-1]
    dp = [[0 for i in range(n + 1)]
          for j in range(m + 1)]
 
    # Fill table in bottom up manner
    for i in range(m + 1):
        for j in range(n + 1):
 
            # Below steps follow recurrence relation
            if i == 0:
                dp[i][j] = j
            elif j == 0:
                dp[i][j] = i
            elif x[i - 1] == y[j - 1]:
                dp[i][j] = 1 + dp[i - 1][j - 1]
            else:
                dp[i][j] = 1 + min(dp[i - 1][j],
                                   dp[i][j - 1])
 
    # Following code is used to print
    # shortest supersequence
 
    # dp[m][n] stores the length of the
    # shortest supersequence of X and Y
 
    # string to store the shortest supersequence
    string = ""
 
    # Start from the bottom right corner and
    # add the characters to the output string
    i = m
    j = n
    while i * j > 0:
 
        # If current character in X and Y are same,
        # then current character is part of
        # shortest supersequence
        if x[i - 1] == y[j - 1]:
 
            # Put current character in result
            string = x[i - 1] + string
 
            # reduce values of i, j and index
            i -= 1
            j -= 1
 
        # If current character in X and Y are different
        elif dp[i - 1][j] > dp[i][j - 1]:
 
            # Put current character of Y in result
            string = y[j - 1] + string
 
            # reduce values of j and index
            j -= 1
        else:
 
            # Put current character of X in result
            string = x[i - 1] + string
 
            # reduce values of i and index
            i -= 1
 
    # If Y reaches its end, put remaining characters
    # of X in the result string
    while i > 0:
        string = x[i - 1] + string
        i -= 1
 
    # If X reaches its end, put remaining characters
    # of Y in the result string
    while j > 0:
        string = y[j - 1] + string
        j -= 1
 
    return string
 
 
# Driver Code
if __name__ == "__main__":
    x = "GXTXAYB"
    y = "AGGTAB"
    m = len(x)
    n = len(y)
 
    # Take the smaller string as x and larger one as y
    if m > n:
        x, y = y, x
        m, n = n, m
 
    print(*printShortestSuperSeq(m, n, x, y))
 
# This code is contributed by
# sanjeev2552

                    

C#

/* A dynamic programming based C# program print
shortest supersequence of two strings */
using System;
 
class GFG
{
 
    // returns shortest supersequence of X and Y
    static String printShortestSuperSeq(String X, String Y)
    {
        int m = X.Length;
        int n = Y.Length;
 
        // dp[i,j] contains length of
        // shortest supersequence
        // for X[0..i-1] and Y[0..j-1]
        int [,]dp = new int[m + 1, n + 1];
        int i, j;
         
        // Fill table in bottom up manner
        for (i = 0; i <= m; i++)
        {
            for (j = 0; j <= n; j++)
            {
                 
                // Below steps follow recurrence relation
                if (i == 0)
                {
                    dp[i, j] = j;
                }
                else if (j == 0)
                {
                    dp[i, j] = i;
                }
                else if (X[i - 1] == Y[j - 1])
                {
                    dp[i, j] = 1 + dp[i - 1, j - 1];
                }
                else
                {
                    dp[i, j] = 1 + Math.Min(dp[i - 1, j], dp[i, j - 1]);
                }
            }
        }
 
        // Following code is used to print
        // shortest supersequence dp[m,n] s
        // tores the length of the shortest
        // supersequence of X and Y
 
        // string to store the shortest supersequence
        String str = "";
 
        // Start from the bottom right corner and one by one
        // push characters in output string
        i = m; j = n;
        while (i > 0 && j > 0)
         
        {
            // If current character in X and Y are same, then
            // current character is part of shortest supersequence
            if (X[i - 1] == Y[j - 1])
             
            {
                // Put current character in result
                str += (X[i - 1]);
 
                // reduce values of i, j and index
                i--;
                j--;
            }
             
            // If current character in X and Y are different
            else if (dp[i - 1, j] > dp[i, j - 1])
            {
                 
                // Put current character of Y in result
                str += (Y[j - 1]);
 
                // reduce values of j and index
                j--;
            }
            else
            {
                 
                // Put current character of X in result
                str += (X[i - 1]);
 
                // reduce values of i and index
                i--;
            }
        }
 
        // If Y reaches its end, put remaining characters
        // of X in the result string
        while (i > 0)
        {
            str += (X[i - 1]);
            i--;
        }
 
        // If X reaches its end, put remaining characters
        // of Y in the result string
        while (j > 0)
        {
            str += (Y[j - 1]);
            j--;
        }
 
        // reverse the string and return it
        str = reverse(str);
        return str;
    }
 
    static String reverse(String input)
    {
        char[] temparray = input.ToCharArray();
        int left, right = 0;
        right = temparray.Length - 1;
 
        for (left = 0; left < right; left++, right--)
        {
            // Swap values of left and right
            char temp = temparray[left];
            temparray[left] = temparray[right];
            temparray[right] = temp;
        }
        return String.Join("",temparray);
    }
     
    // Driver code
    public static void Main(String[] args)
    {
        String X = "AGGTAB";
        String Y = "GXTXAYB";
        Console.WriteLine(printShortestSuperSeq(X, Y));
    }
}
 
/* This code has been contributed
by PrinciRaj1992*/

                    

Javascript

<script>
 
/* A dynamic programming based Javascript program print
shortest supersequence of two strings */
 
// returns shortest supersequence of X and Y
function printShortestSuperSeq(X,Y)
{
         let m = X.length;
        let n = Y.length;
   
        // dp[i][j] contains length of
        // shortest supersequence
        // for X[0..i-1] and Y[0..j-1]
        let dp = new Array(m + 1);
        for(let i=0;i<(m+1);i++)
        {
            dp[i]=new Array(n+1);
            for(let j=0;j<(n+1);j++)
                dp[i][j]=0;
        }
   
        // Fill table in bottom up manner
        for (let i = 0; i <= m; i++)
        {
            for (let j = 0; j <= n; j++)
            {
                   
                // Below steps follow recurrence relation
                if (i == 0)
                {
                    dp[i][j] = j;
                }
                else if (j == 0)
                {
                    dp[i][j] = i;
                }
                else if (X[i-1] == Y[j-1])
                {
                    dp[i][j] = 1 + dp[i - 1][j - 1];
                }
                else
                {
                    dp[i][j] =
                    1 + Math.min(dp[i - 1][j], dp[i][j - 1]);
                }
            }
        }
   
        // Following code is used to print
        // shortest supersequence dp[m][n] s
        // tores the length of the shortest
        // supersequence of X and Y
   
        // string to store the shortest supersequence
        let str = "";
   
        // Start from the bottom right corner and one by one
        // push characters in output string
        let i = m, j = n;
        while (i > 0 && j > 0)
           
        {
            // If current character in X and Y are same, then
            // current character is part of shortest supersequence
            if (X[i-1] == Y[j-1])
               
            {
                // Put current character in result
                str += (X[i-1]);
   
                // reduce values of i, j and index
                i--;
                j--;
            }
               
            // If current character in X and Y are different
            else if (dp[i - 1][j] > dp[i][j - 1])
            {
                   
                // Put current character of Y in result
                str += (Y[j-1]);
   
                // reduce values of j and index
                j--;
            }
            else
            {
                   
                // Put current character of X in result
                str += (X[i-1]);
   
                // reduce values of i and index
                i--;
            }
        }
   
        // If Y reaches its end, put remaining characters
        // of X in the result string
        while (i > 0)
        {
            str += (X[i-1]);
            i--;
        }
   
        // If X reaches its end, put remaining characters
        // of Y in the result string
        while (j > 0)
        {
            str += (Y[j-1]);
            j--;
        }
   
        // reverse the string and return it
        str = reverse(str);
        return str;
}
 
function reverse(input)
{
    let temparray = input.split("");
        let left, right = 0;
        right = temparray.length - 1;
   
        for (left = 0; left < right; left++, right--)
        {
            // Swap values of left and right
            let temp = temparray[left];
            temparray[left] = temparray[right];
            temparray[right] = temp;
        }
        return (temparray).join("");
}
 
// Driver code
let X = "AGGTAB";
let Y = "GXTXAYB";
document.write(printShortestSuperSeq(X, Y));
 
 
// This code is contributed by rag2127
 
</script>

                    

Output
AGXGTXAYB


Time complexity of above solution is O(n2). 
Auxiliary space used by the program is O(n2).


 



Last Updated : 02 Jan, 2024
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