### Problem Statement

Given two words

*word1*and*word2*, find the minimum number of operations required to convert*word1*to*word2*.
You have the following 3 operations permitted on a word:

- Insert a character
- Delete a character
- Replace a character

**Example 1:**

Input:word1 = "horse", word2 = "ros"Output:3Explanation:horse -> rorse (replace 'h' with 'r') rorse -> rose (remove 'r') rose -> ros (remove 'e')

**Example 2:**

Problem linkInput:word1 = "intention", word2 = "execution"Output:5Explanation:intention -> inention (remove 't') inention -> enention (replace 'i' with 'e') enention -> exention (replace 'n' with 'x') exention -> exection (replace 'n' with 'c') exection -> execution (insert 'u')

### Video Tutorial

You can find the detailed video tutorial here### Thought Process

Brute force sounds like really overkill because we have to list all the possible ways and it is exponential time complexity. On a high level, the recursion looks like- if cannot convert A to B, return -1 (the recursion termination function); if A == B, then return 0
- try 3 different ways, insert, remove and replace, cut that character and continue the recursion. Compare the minimum of that 3 methods (exclude -1)

The mathematical induction function is

DP[i][j] is the minimum operations needed to convert String[0-i] to String[0-j]

Initial values: dp[0][j] = j and dp[i][0] = i

If (A[i] == B[j]) dp[i][j] = dp[i-1][j-1]

Else min(dp[i][j-1], dp[i-1][j], dp[i-1][j-1]) +1;

### Solutions

#### DP

Time Complexity: O(M*N) where M is word1 length and N is word2 lengthSpace Complexity: O(M*N) since we need an extra 2D array

References

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