Leetcode solution 72. Edit Distance

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:

1. Insert a character
2. Delete a character
3. Replace a character

Example 1:

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

Example 2:

Input: word1 = "intention", word2 = "execution"
Output: 5
Explanation: 
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')

 Problem link

Video Tutorial

You can find the detailed video tutorial here

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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)

Edit distance is a classic Dynamic Programming (DP) problem, just like Coin Change Problem. It follows the DP template perfectly (asking for extreme values)

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 length
Space Complexity: O(M*N) since we need an extra 2D array

References

• Leetcode official solution