Problem Description

The problem entails figuring out the least number of steps necessary to equalize two strings, word1 and word2. A single step consists of deleting one character from either one of the strings. This procedure is repeated until both strings are identical. The goal is to identify the minimum number of such deletions to make the two strings match.

Intuition

The intuition behind solving this task lies in the concept of finding the longest common subsequence (LCS) between the two strings. The LCS is the longest sequence of characters that appear in both strings in the same order, possibly with other characters in between. Once we find the LCS, we know that the characters not part of the LCS need to be deleted to make the strings the same.

To find the LCS, we can use dynamic programming. The main idea is to construct a 2D array dp where each element dp[i][j] represents the length of the LCS between word1 up to index i and word2 up to index j. If the characters at word1[i] and word2[j] match, it means we can extend the LCS by one more character, so we take the LCS up to word1[i-1] and word2[j-1]. If they do not match, we take the longer of the LCSs without the current character of either word1 or word2. The edges of the array are initialized to the index values, representing the need to delete all previous characters in a string to match an empty string.

Once the dp array is fully populated, the value at dp[m][n] (where m and n are the lengths of word1 and word2, respectively) will give us the length of the LCS. The minimum number of deletions required is then the sum of the lengths of the two input strings minus twice the length of the LCS, which accounts for every character not in the LCS needing to be deleted from both strings.

Learn more about Dynamic Programming patterns.

Solution Approach

The solution adopts a dynamic programming approach to efficiently compute the number of necessary deletion steps. Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems and storing the results of those subproblems to avoid redundant computations.

Here is the breakdown of the implementation:

1. Initialize the 2D array dp with dimensions (m+1) x (n+1), where m is the length of word1 and n is the length of word2. This array will hold the lengths of LCS for different substrings of word1 and word2. We add 1 to include the empty string cases as well.

2. Fill the first row and first column of dp with increasing indices. This takes into account that converting any string to an empty string requires a number of deletions equal to the length of the string.

3. Loop through the matrix starting at dp[1][1] and decide for each pair (i, j):

   • If word1[i - 1] is equal to word2[j - 1], then dp[i][j] is set to dp[i - 1][j - 1] as the characters match and do not need to be deleted (we extend the LCS).
   • If the characters do not match, we look at two scenarios: deleting the last character from word1 or word2. The value of dp[i][j] is set to the minimum of the two possibilities plus one (for the deletion step): dp[i][j] = 1 + min(dp[i - 1][j], dp[i][j - 1]).

4. After completely filling in the dp table, the value at dp[m][n] is the minimum number of deletion steps necessary for the two words to become the same. This is because dp[m][n] represents the size of the LCS, and by subtracting this from the total length of the two strings, we get the number of characters not part of the LCS, which is exactly the number of deletions needed.

5. Return dp[m][n] as the solution.

The algorithm uses a 2D array for dynamic programming, which is a common data structure for storing intermediary results in dynamic programming tasks. The pattern used is to build up the solution from the smallest subproblems (empty strings) to the full problem by adding one character at a time and determining what the best decision is at each step.

Example Walkthrough

Let's walk through the solution approach with a small example where word1 = "sea" and word2 = "eat". The goal is to find the minimum number of deletions to make word1 equal to word2.

1. Initialize the 2D array dp. Given that m = 3 (the length of "sea") and n = 3 (the length of "eat"), the dp array will have dimensions (4 x 4) to include the empty string cases.

2. Fill the first row and column with increasing indices, reflecting the deletion count to match an empty string.

   dp = [
       [0, 1, 2, 3],  // "" -> ""
       [1, 0, 0, 0],  // "s" -> ""
       [2, 0, 0, 0],  // "se" -> ""
       [3, 0, 0, 0]   // "sea" -> ""
   ]

3. Loop through the dp array and fill it following the rules:

   • If word1[i - 1] equals word2[j - 1], set dp[i][j] to dp[i - 1][j - 1].
   • If they do not match, set dp[i][j] to the minimum of dp[i - 1][j] and dp[i][j - 1] plus one.
   • Following this logic, the array is updated as shown below (explanations in comments):

   dp = [
       [0, 1, 2, 3],                // Starting row and column
       [1, 1, 2, 2],                // "s" -> "e" or "" -> "e", "s" -> "ea"
       [2, 2, 1, 2],                // "se" -> "e", "se" -> "ea" or "s" -> "ea"
       [3, 2, 2, (1 for 'ea')]     // "sea" -> "e" or "s" -> "e", "sea" -> "ea"  : 'a' matches
   ]

   The last cell is '1' since 'a' at the end of both words matches, so we carry over the length of LCS without 'a', which was 0 (since "se" -> "e" required no LCS extensions), and add 1 for the common 'a'.

4. After filling in the dp table, the last cell dp[3][3] now contains the length of the LCS. This represents that the strings can become the same with a minimum of 2 deletion steps (m + n - 2 * dp[3][3] = 3 + 3 - 2 * 1 = 4).

5. The answer is 4, which means that 4 deletions are necessary for word1 and word2 to become the same.

In conclusion, using dynamic programming, the algorithm iteratively finds the longest common subsequence and uses its length to calculate the minimal number of deletions required to equalize the provided strings.

Solution Implementation

Time and Space Complexity

The given Python code implements the solution to find the minimum number of operations needed to convert word1 to word2, where the operations can be insertions, deletions, or substitutions of characters.

Time Complexity

The time complexity of the code can be analyzed based on the nested for loops:

• The outer loop runs m times, where m is the length of word1.
• The inner loop runs n times for each iteration of the outer loop, where n is the length of word2.

Because each cell in the dp array is computed once, the total number of computations is proportional to the product of m and n. Therefore, the time complexity of the code is O(m * n).

Space Complexity

The space complexity is determined by the size of the dp array used in the dynamic programming approach:

• The dp array is a 2D array with dimensions (m + 1) x (n + 1).

Therefore, the amount of space used is proportional to the product of (m + 1) and (n + 1). Though the +1 is constant and does not affect the asymptotic complexity, it accounts for the extra row and column needed for the base cases. Consequently, the space complexity of the code is also O(m * n).

Learn more about how to find time and space complexity quickly using problem constraints.