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Minimum Edit Distance - Dyanamic Programming

Problem Statement : - 

  • Given two strings str1 and str2 and below operations that can performed on str1. Find minimum number of edits (operations) required to convert ‘str1’ into ‘str2’.
  • Operations and their Cost which we can perform to achieve this goal
    • Insert - 1
    • Delete - 1
    • Replace - 2 (Note : Delete The Element + Insert New Element)
Example 1 : -
  Input       : s1= Hogward , s2= Hogwartz
  Output      : 4
  Explanation : 1. Insert 'g' at index 3 ; cost = 1
                2. Replace 'd' with 't' at index 7; cost = 2
                3. Insert 'z' at index 8 = 1; cost=1
                Total cost = 1+2+1 = 4
Idea : -
  1. We create array MED of  Size M * N where M= length of s1 N= length of s2
  2. To Solve This Problem We use Dynamic Approach.
  3. We solve All sub problem which is |sub1|=m , |sub2|=n  where m <=|s1| and n<=|s2|
  4. If last characters of two strings are same, nothing need to do. So we use Previous Solution and Recur for  (MED(i-1,j-1)).
  5. Else (If last characters are not same), we consider all operations on ‘str1’, consider all three operations on last character of first string, recursively compute minimum cost for all three operations and take minimum of three values.
    1. Insert: Recur for MED(m, n-1)
    2. Remove: Recur for MED(m-1, n)
    3. Replace: Recur for MED(m-1, n-1)
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Base Conditions : -
  • for |s1|=0 and |s2|=0 
    • MED[0][0]=0
Sub Problems : -
    

  • for |s1|=0 and |s2| > 0 we required to insert each element of s2 in s1.
    • 

      

    • MED[0][1]=1
      • 

    • MED[0][2]=2
      • 

    

  • for |s2|=0 and |s1| > 0 we required to delete each element of s1.
    • 

      

    • MED[1][0]=1
      • 

    • MED[2][0]=2
      • 

    

  • otherwise
    • 

      

    • MED[i][j]=min{MED[i-1][j]+1,MED[i][j-1]+1,MED[i-1][j-1]+2}   if(s1[i]!=s2[i])
      • 

    • MED[i][j]=MED[i-1][j-1]   if(S1[i]=s2[j])
      •
Example 2  :-
  S1 = ALIBABA
  S2 = ALLADIN
MED Array : -

 
 


  
  
  
  
  
  
  
  
  
  
 



  
  S2
  ^
  A
  L
  I
  B
  A
  B
  A
 



          
  
  
  
  
  
  
  
  
  
 



  S1
  i\j
  0   
  1
  2
  3
  4
  5
  6
  7
 



  ^
  0       
  0    
  1
  2
  3
  4
  5
  6
  7
 



  A
  1
  1
  0
  1
  2
  3
  4
  5
  6
 



  L
  2
  2
  1
  0
  1
  2
  3
  4
  5
 



  L
  3
  3
  2
  1
  2
  3
  4
  5
  6
 



  A
  4
  4
  3
  2
  3
  4
  3
  4
  5
 



  D
  5
  5
  4
  3
  4
  5
  4
  5
  6
 



  I
  6
  6
  5
  4
  3
  4
  5
  6
  7
 



  N
  7
  7
  6
  5
  4
  5
  6
  7
  8

Algo : -
for(j=0 to M)
   MED[0][j]=j
for(i=0 to N)
   MED[i][0]=i;
for(i=1 to N)
   for(j=1 to M)
      if(s1[i]==s2[i])
        MED[i][j]=MED[i-1][j-1]
      else
        MED[i][j]=min{MED[i-1][j]+1,MED[i][j-1]+1,MED[i-1][j-1]+2}
Programme : -
Download Programme
#include<stdio.h>
#include<string.h>
int min(int a,int b,int c)
{
    if(a<b)
    {
        return(c<a?c:a);
    }
    else
    {
        return(c<b?c:b);
    }
}
main()
{
    int i,j;
    char s1[20],s2[20];
    printf("ENTER THE FIRST STRING : -");
    scanf("%s",s1);
    printf("ENTER THE SECOND STRING : -");
    scanf("%s",s2);
    int M,N;
    M=strlen(s1);
    N=strlen(s2);
    int MED[M+1][N+1];
    for(i=0;i<=N;i++)
    {
        MED[i][0]=i;
    }
    for(j=0;j<=M;j++)
    {
        MED[0][j]=j;
    }
    for(i=1;i<=N;i++)
    {
        for(j=1;j<=M;j++)
        {
            if(s1[i-1]==s2[j-1])
            {
                MED[i][j]=MED[i-1][j-1];
            }
            else
            {
                MED[i][j]=min((MED[i-1][j]+1),(MED[i][j-1]+1),(MED[i-1][j-1]+2));
            }

        }

    }
    printf("   ^  ");
    for(i=0;i<M;i++)
    {
        printf("%c  ",s2[i]);
    }
    printf("\n");
    printf("^  ");
    for(i=0;i<=N;i++)
    {

        for(j=0;j<=M;j++)
        {
            printf("%d  ",MED[i][j]);
        }

         printf("\n");
            printf("%c  ",s1[i]);
    }

    printf("\nCOST OF CONVERTING '%s' TO '%s' IS  : %d",s1,s2,MED[N][M]);

}


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