The Edit Distance Problem, also known as the Levenshtein distance, measures how dissimilar two strings are by counting the minimum number of operations required to transform one string into another. These operations typically include insertions, deletions, and substitutions of characters. Understanding this problem is crucial not only for coding interviews but also for applications like spell checking, DNA sequencing, and natural language processing.

Understanding the Problem

Let's break it down with a simple example. Consider the two strings:

• s1: "kitten"
• s2: "sitting"

To transform "kitten" into "sitting", our goal is to find the minimum number of edits needed. Here’s one possible sequence of operations:

1. Substitute 'k' with 's'
2. Substitute 'e' with 'i'
3. Insert 'g' at the end

Thus, the edit distance between "kitten" and "sitting" is 3.

Operations Defined

To clarify, here are the three operations:

1. Insert: Add a character in string s1.
2. Remove: Delete a character from string s1.
3. Replace: Change a character in string s1 to match the corresponding character in string s2.

Dynamic Programming Approach

To efficiently calculate the edit distance, we can use a dynamic programming approach. The idea is to construct a 2D array (or matrix) where the cell dp[i][j] represents the edit distance between the first i characters of s1 and the first j characters of s2.

Step-by-Step Construction of the DP Matrix

1. Initialization:

   • Create a matrix dp of size (m+1) x (n+1), where m is the length of s1 and n is the length of s2.
   • Initialize the first row and column:
     • dp[i][0] = i: it takes i deletions to convert the first i characters of s1 to an empty string.
     • dp[0][j] = j: it takes j insertions to convert an empty string to the first j characters of s2.

2. Filling the Matrix:

   • For each i (from 1 to m) and j (from 1 to n):
     • If the characters match (i.e., s1[i-1] == s2[j-1]), then no additional edit is needed:
       • dp[i][j] = dp[i-1][j-1]
     • If they don't match, consider the minimum of the three possible operations:
       • Insert: dp[i][j-1] + 1
       • Remove: dp[i-1][j] + 1
       • Replace: dp[i-1][j-1] + 1
     • Thus, the formula becomes:

       dp[i][j] = min(dp[i-1][j] + 1,   // Delete
                      dp[i][j-1] + 1,   // Insert
                      dp[i-1][j-1] + 1) // Replace
       

3. Result:

   • After filling the matrix, the value dp[m][n] will contain the minimum edit distance between the two strings.

Example Implementation

Here's a basic Python implementation of the above approach:

def edit_distance(s1, s2):
    m = len(s1)
    n = len(s2)
    
    # Create a DP table
    dp = [[0 for _ in range(n + 1)] for _ in range(m + 1)]
    
    # Initialize base cases
    for i in range(m + 1):
        dp[i][0] = i
    for j in range(n + 1):
        dp[0][j] = j

    # Compute distances
    for i in range(1, m + 1):
        for j in range(1, n + 1):
            if s1[i - 1] == s2[j - 1]:
                dp[i][j] = dp[i - 1][j - 1]
            else:
                dp[i][j] = min(dp[i - 1][j] + 1,    # Deletion
                               dp[i][j - 1] + 1,    # Insertion
                               dp[i - 1][j - 1] + 1) # Replacement

    return dp[m][n]

# Example usage:
s1 = "kitten"
s2 = "sitting"
print(f"Edit Distance: {edit_distance(s1, s2)}")  # Output: 3

Time and Space Complexity

The algorithm runs with a time complexity of (O(m \times n)), where (m) and (n) are the lengths of the two strings. Similarly, the space complexity is also (O(m \times n)) due to the DP matrix. However, we can optimize the space usage to (O(\min(m, n))) by only keeping track of the current and previous rows.

Understanding the Edit Distance Problem will enhance your problem-solving skills in dynamic programming, especially for a range of applications beyond string comparison, such as data transformation and error correction in coding systems.