# 1143. Longest Common Subsequence

## Description

Given two strings text1 and text2, return the length of their longest common subsequence. If there is no common subsequence, return 0.

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.

• For example, "ace" is a subsequence of "abcde".

A common subsequence of two strings is a subsequence that is common to both strings.

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.

Constraints:

• 1 <= text1.length, text2.length <= 1000
• text1 and text2 consist of only lowercase English characters.

## Solutions

Solution 1: Dynamic Programming

We define $f[i][j]$ as the length of the longest common subsequence of the first $i$ characters of $text1$ and the first $j$ characters of $text2$. Therefore, the answer is $f[m][n]$, where $m$ and $n$ are the lengths of $text1$ and $text2$, respectively.

If the $i$th character of $text1$ and the $j$th character of $text2$ are the same, then $f[i][j] = f[i - 1][j - 1] + 1$; if the $i$th character of $text1$ and the $j$th character of $text2$ are different, then $f[i][j] = max(f[i - 1][j], f[i][j - 1])$. The state transition equation is:

$f[i][j] = \begin{cases} f[i - 1][j - 1] + 1, & \text{if } text1[i - 1] = text2[j - 1] \\ \max(f[i - 1][j], f[i][j - 1]), & \text{if } text1[i - 1] \neq text2[j - 1] \end{cases}$

The time complexity is $O(m \times n)$, and the space complexity is $O(m \times n)$. Here, $m$ and $n$ are the lengths of $text1$ and $text2$, respectively.

• class Solution {
public int longestCommonSubsequence(String text1, String text2) {
int m = text1.length(), n = text2.length();
int[][] f = 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)) {
f[i][j] = f[i - 1][j - 1] + 1;
} else {
f[i][j] = Math.max(f[i - 1][j], f[i][j - 1]);
}
}
}
return f[m][n];
}
}

• class Solution {
public:
int longestCommonSubsequence(string text1, string text2) {
int m = text1.size(), n = text2.size();
int f[m + 1][n + 1];
memset(f, 0, sizeof f);
for (int i = 1; i <= m; ++i) {
for (int j = 1; j <= n; ++j) {
if (text1[i - 1] == text2[j - 1]) {
f[i][j] = f[i - 1][j - 1] + 1;
} else {
f[i][j] = max(f[i - 1][j], f[i][j - 1]);
}
}
}
return f[m][n];
}
};

• class Solution:
def longestCommonSubsequence(self, text1: str, text2: str) -> int:
m, n = len(text1), len(text2)
f = [[0] * (n + 1) for _ in range(m + 1)]
for i in range(1, m + 1):
for j in range(1, n + 1):
if text1[i - 1] == text2[j - 1]:
f[i][j] = f[i - 1][j - 1] + 1
else:
f[i][j] = max(f[i - 1][j], f[i][j - 1])
return f[m][n]

• func longestCommonSubsequence(text1 string, text2 string) int {
m, n := len(text1), len(text2)
f := make([][]int, m+1)
for i := range f {
f[i] = make([]int, n+1)
}
for i := 1; i <= m; i++ {
for j := 1; j <= n; j++ {
if text1[i-1] == text2[j-1] {
f[i][j] = f[i-1][j-1] + 1
} else {
f[i][j] = max(f[i-1][j], f[i][j-1])
}
}
}
return f[m][n]
}

• function longestCommonSubsequence(text1: string, text2: string): number {
const m = text1.length;
const n = text2.length;
const f = Array.from({ length: m + 1 }, () => Array(n + 1).fill(0));
for (let i = 1; i <= m; i++) {
for (let j = 1; j <= n; j++) {
if (text1[i - 1] === text2[j - 1]) {
f[i][j] = f[i - 1][j - 1] + 1;
} else {
f[i][j] = Math.max(f[i - 1][j], f[i][j - 1]);
}
}
}
return f[m][n];
}

• /**
* @param {string} text1
* @param {string} text2
* @return {number}
*/
var longestCommonSubsequence = function (text1, text2) {
const m = text1.length;
const n = text2.length;
const f = Array.from({ length: m + 1 }, () => Array(n + 1).fill(0));
for (let i = 1; i <= m; ++i) {
for (let j = 1; j <= n; ++j) {
if (text1[i - 1] == text2[j - 1]) {
f[i][j] = f[i - 1][j - 1] + 1;
} else {
f[i][j] = Math.max(f[i - 1][j], f[i][j - 1]);
}
}
}
return f[m][n];
};

• public class Solution {
public int LongestCommonSubsequence(string text1, string text2) {
int m = text1.Length, n = text2.Length;
int[,] f = new int[m + 1, n + 1];
for (int i = 1; i <= m; ++i) {
for (int j = 1; j <= n; ++j) {
if (text1[i - 1] == text2[j - 1]) {
f[i, j] = f[i - 1, j - 1] + 1;
} else {
f[i, j] = Math.Max(f[i - 1, j], f[i, j - 1]);
}
}
}
return f[m, n];
}
}

• impl Solution {
pub fn longest_common_subsequence(text1: String, text2: String) -> i32 {
let (m, n) = (text1.len(), text2.len());
let (text1, text2) = (text1.as_bytes(), text2.as_bytes());
let mut f = vec![vec![0; n + 1]; m + 1];
for i in 1..=m {
for j in 1..=n {
f[i][j] = if text1[i - 1] == text2[j - 1] {
f[i - 1][j - 1] + 1
} else {
f[i - 1][j].max(f[i][j - 1])
};
}
}
f[m][n]
}
}