Edit distance and LCS (Longest Common Subsequence)

In standard Edit Distance where we are allowed 3 operations, insert, delete and replace. Consider a variation of edit distance where we are allowed only two operations insert and delete, find edit distance in this variation.
Examples:

Input : str1 = "cat", st2 = "cut"
Output : 2
We are allowed to insert and delete. We delete 'a'
from "cat" and insert "u" to make it "cut".

Input : str1 = "acb", st2 = "ab"
Output : 1
We can convert "acb" to "ab" by removing 'c'.

One solution is to simply modify Edit Distance Solution by making two recursive call instead of three. An interesting solution is based on LCS.
1) Find LCS of two strings. Let length of LCS be x.
2) Let length of first string be m and length of second string be n. Our result is (m – x) + (n – x). We basically need to do (m – x) delete operations and (n – x) insert operations.

C++

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// CPP program to find Edit Distance (when only two
// operations are allowed, insert and delete) using LCS.
#include<bits/stdc++.h>
using namespace std;
  
int editDistanceWith2Ops(string &X, string &Y)
{
    // Find LCS
    int m = X.length(), n = Y.length(); 
    int L[m+1][n+1]; 
    for (int i=0; i<=m; i++)
    {
        for (int j=0; j<=n; j++)
        {
            if (i == 0 || j == 0)
                L[i][j] = 0;
            else if (X[i-1] == Y[j-1])
                L[i][j] = L[i-1][j-1] + 1;
            else
                L[i][j] = max(L[i-1][j], L[i][j-1]);
        }
    }    
    int lcs  = L[m][n];
  
    // Edit distance is delete operations + 
    // insert operations.
    return (m - lcs) + (n - lcs);
}
  
/* Driver program to test above function */
int main()
{
    string X = "abc", Y = "acd";
    cout << editDistanceWith2Ops(X, Y);
    return 0;
}

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Java

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//Java program to find Edit Distance (when only two 
// operations are allowed, insert and delete) using LCS.
  
class GFG {
  
    static int editDistanceWith2Ops(String X, String Y) {
        // Find LCS 
        int m = X.length(), n = Y.length();
        int L[][] = new int[m + 1][n + 1];
        for (int i = 0; i <= m; i++) {
            for (int j = 0; j <= n; j++) {
                if (i == 0 || j == 0) {
                    L[i][j] = 0;
                } else if (X.charAt(i - 1) == Y.charAt(j - 1)) {
                    L[i][j] = L[i - 1][j - 1] + 1;
                } else {
                    L[i][j] = Math.max(L[i - 1][j], L[i][j - 1]);
                }
            }
        }
        int lcs = L[m][n];
  
        // Edit distance is delete operations + 
        // insert operations. 
        return (m - lcs) + (n - lcs);
    }
  
    /* Driver program to test above function */
    public static void main(String[] args) {
        String X = "abc", Y = "acd";
        System.out.println(editDistanceWith2Ops(X, Y));
  
    }
}
/* This Java code is contributed by 29AjayKumar*/

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Python 3

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# Python 3 program to find Edit Distance 
# (when only two operations are allowed,
# insert and delete) using LCS.
  
def editDistanceWith2Ops(X, Y):
  
    # Find LCS
    m = len(X) 
    n = len(Y)
    L = [[0 for x in range(m + 1)] 
            for y in range(n + 1)] 
    for i in range(m + 1):
        for j in range(n + 1):
            if (i == 0 or j == 0):
                L[i][j] = 0
            elif (X[i - 1] == Y[j - 1]):
                L[i][j] = L[i - 1][j - 1] + 1
            else:
                L[i][j] = max(L[i - 1][j],
                              L[i][j - 1])
  
    lcs = L[m][n]
  
    # Edit distance is delete operations + 
    # insert operations.
    return (m - lcs) + (n - lcs)
  
# Driver Code
if __name__ == "__main__":
      
    X = "abc"
    Y = "acd"
    print(editDistanceWith2Ops(X, Y))
  
# This code is contributed by ita_c

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C#

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// C# program to find Edit Distance 
// (when only two operations are 
// allowed, insert and delete) using LCS. 
using System;
  
class GFG 
  
static int editDistanceWith2Ops(String X, 
                                String Y) 
    // Find LCS 
    int m = X.Length, n = Y.Length; 
    int [ , ]L = new int[m + 1 , n + 1]; 
    for (int i = 0; i <= m; i++)
    
        for (int j = 0; j <= n; j++) 
        
            if (i == 0 || j == 0)
            
                L[i , j] = 0; 
            
            else if (X[i - 1] == Y[j - 1]) 
            
                L[i , j] = L[i - 1 , j - 1] + 1; 
            
            else 
            
                L[i , j] = Math.Max(L[i - 1 , j], 
                                    L[i , j - 1]); 
            
        
    
    int lcs = L[m , n]; 
  
    // Edit distance is delete operations + 
    // insert operations. 
    return (m - lcs) + (n - lcs); 
  
// Driver Code
public static void Main() 
    String X = "abc", Y = "acd"
    Console.Write(editDistanceWith2Ops(X, Y)); 
  
// This code is contributed 
// by 29AjayKumar

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PHP

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<?php
// PHP program to find Edit Distance 
// (when only two operations are allowed,
// insert and delete) using LCS.
  
function editDistanceWith2Ops($X, $Y)
{
    // Find LCS
    $m = strlen($X); $n = strlen($Y); 
    $L[$m + 1][$n + 1]; 
    for ($i = 0; $i <= $m; $i++)
    {
        for ($j = 0; $j <= $n; $j++)
        {
            if ($i == 0 || $j == 0)
                $L[$i][$j] = 0;
            else if ($X[$i - 1] == $Y[$j - 1])
                $L[$i][$j] = $L[$i - 1][$j - 1] + 1;
            else
                $L[$i][$j] = max($L[$i - 1][$j], 
                                 $L[$i][$j - 1]);
        }
    
    $lcs = $L[$m][$n];
  
    // Edit distance is delete operations + 
    // insert operations.
    return ($m - $lcs) + ($n - $lcs);
}
  
// Driver Code
$X = "abc"; $Y = "acd";
echo editDistanceWith2Ops($X, $Y);
  
// This code is contributed
// by Akanksha Rai
?>

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Output:



2

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



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