## Problem

Problem ID: 1143

Title: Longest Common Subsequence

Description: Given two strings, find the length of the longest common subsequence.

## Thoughts

The first thing that came to my mind after reading this question is DP as it involved comparing 2 different strings. However, I was having trouble with deriving the DP state transition as it involved.

My initial approach was to store a 4D DP where DP[i_l][i_r][j_l][j_r] will store the longest length for text1[i_l][i_r] and text[j_l][j_r] but this would result in TLE as text1.length < 1000 and the total run time would be O(n^4) = 1e12. I soon realise that we should greedily add characters to the subsequence and there is no need to store the left bound in the DP which would reduce the complexity to O(n^2).

## Solution

### Explanation

General Idea: Lets say you are currently at index i and j of text1 and text2. There are three scenario you should consider to calculate the longest subsequence up till i and j.

1. What is the longest subsequence up till i-1 and j. The similarity of characters text1[i] and text2[j] will not contribute to longest subsequence as text2[j] has already been used.
2. What is the longest subsequence up till i and j-1. The similarity of characters text1[i] and text2[j] will not contribute to longest subsequence as text1[i] has already been used.
3. What is the longest subsequence up till i-1 and j-1. The similarity of text1[i] and text2[j] can contribute to the longest subsequence.

Using this idea we can formulate the following DP transition.

DP

DP[i][j]: stores the longest subsequence for the first i characters of text1 and first j characters of text2. Even though I prefer to use index of characters instead of length for DP parameters, I use length for this problem to allow us to not check for index in range.

DP transition: DP[i][j] = max(DP[i-1][j-1], DP[i][j-1], DP[i-1][j-1] + text1[i-1] == text2[j-1] ? 1 : 0)

Implementation

class Solution {
public:
int longestCommonSubsequence(string text1, string text2) {
int n = text1.size();
int m = text2.size();
vector<vector<int>> dp(n + 1, vector<int>(m + 1, 0));
for (int i = 1; i <= n; i++) {
for (int j = 1; j <= m; j++) {
char c1 = text1[i - 1];
char c2 = text2[j - 1];
int exclude_curr = max(dp[i-1][j], dp[i][j-1]);
int curr_take = dp[i-1][j-1];
if (c1 == c2) curr_take++;
dp[i][j] = max(curr_take, exclude_curr);
}
}
return dp[n][m];
}
};