# longest-common-subsequect

## Problem

[Longest Common Subsequece](https://leetcode.com/explore/other/card/30-day-leetcoding-challenge/531/week-4/3311/)

## Problem Description

```
Given two strings text1 and text2, return the length of their longest common subsequence.

A subsequence of a string is a new string generated from the original string with some characters(can be none) deleted without changing the relative order of the remaining characters. (eg, "ace" is a subsequence of "abcde" while "aec" is not). A common subsequence of two strings is a subsequence that is common to both strings.

If there is no common subsequence, return 0.

Example 1:

Input: text1 = "abcde", text2 = "ace" 
Output: 3  
Explanation: The longest common subsequence is "ace" and its length is 3.

Example 2:

Input: text1 = "abc", text2 = "abc"
Output: 3
Explanation: The longest common subsequence is "abc" and its length is 3.

Example 3:

Input: text1 = "abc", text2 = "def"
Output: 0
Explanation: There is no such common subsequence, so the result is 0.
```

## Solution

This is LCS, a Dynamic Programming (DP) problem, which can break down into smaller, simpler subproblems, and so on. DP problem usually reuse solutions to lower level subproblems.

We define `dp[m+1][n+1]` represents a set of longest common subsequect of prefiex Xi and Yj. given that:

* dp\[0]\[0] = 0;
* if X\[i - 1]==Y\[j -1] (current character is the same), dp\[i]\[j] = 1 + d\[i - 1]\[j - 1]
* if X\[i-1]!=Y\[j-1], dp\[i]\[j] = max(dp\[i-1],j],dp\[i]\[j-1])

repeat until computed the whole string, dp\[m]\[n] is the answer.

For example:

![Longest Common Subsequence](https://388701358-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LmDY11BLFD0Iupj9U9t%2F-M5tpUlj1lm7qWVPSYnp%2F-M5tpVT9O_-Ug8GLqdQE%2Flongest-common-subsequence.png?generation=1587960745446196\&alt=media)

### Complexity Analysis

**Time Complexity:** `O(N*M)`

**Space Complexity:** `O(N*M)`

* N - the length of text1
* M - the length of text2

### Code

```java
class Solution {
    public int longestCommonSubsequence(String text1, String text2) {
        if (text1.equals(text2)) return text1.length();
        int m = text1.length();
        int n = text2.length();
        int[][] dp = new int[m + 1][n + 1];
        for (int i = 1; i <= m; i++) {
            for (int j = 1; j <= n; j++) {
                if (text1.charAt(i - 1) == text2.charAt(j - 1)) {
                dp[i][j] = 1 + dp[i - 1][j - 1];
            } else {
                dp[i][j] = Math.max(dp[i - 1][j], dp[i][j - 1]);
            }
        }
        return dp[m][n];
    }
}
```

## Reference

* [Longest Common Subsequence Problem Wiki](https://en.wikipedia.org/wiki/Longest_common_subsequence_problem)

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