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:
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.