Edit Distance - Video Solutions

Problem Overview: You are given two strings word1 and word2. The task is to compute the minimum number of operations required to convert word1 into word2. Allowed operations are insert, delete, and replace. This classic problem is known as Levenshtein Distance and heavily relies on dynamic programming over prefixes of the two strings.

Approach 1: Recursive Approach with Memoization (Time: O(m*n), Space: O(m*n))

Think of the problem as comparing characters starting from indices i in word1 and j in word2. If the characters match, move both pointers forward with no cost. If they differ, try the three valid operations: insert (advance j), delete (advance i), or replace (advance both). Each operation adds one step. The naive recursion explores an exponential number of states, but many subproblems repeat. Store results in a memo table keyed by (i, j). Memoization ensures each state is solved once, reducing time to O(m*n). This approach clearly expresses the decision process and works well when learning recursion on string problems.

Approach 2: Dynamic Programming Table (Time: O(m*n), Space: O(m*n))

Build a 2D DP table where dp[i][j] represents the minimum operations needed to convert the first i characters of word1 into the first j characters of word2. Initialize base cases: converting an empty string to length j requires j insertions, while converting length i to empty requires i deletions. Iterate through both strings and compare characters. If word1[i-1] == word2[j-1], carry over dp[i-1][j-1]. Otherwise compute 1 + min(insert, delete, replace) using dp[i][j-1], dp[i-1][j], and dp[i-1][j-1]. This bottom-up method systematically evaluates every prefix combination and avoids recursion overhead. It’s the standard approach for problems involving transformations between two strings.

The DP table can also be optimized to O(min(m,n)) space by storing only the current and previous rows since each state depends on adjacent rows. The time complexity remains O(m*n) because every pair of indices must still be evaluated.

Recommended for interviews: The bottom-up dynamic programming solution is what most interviewers expect. It demonstrates clear understanding of state definition, transitions, and base cases. Starting with the recursive reasoning helps explain the problem structure, but implementing the optimized DP table shows strong command of classic string transformation problems.