[LeetCode] 1143. Longest Common Subsequence 最長公共子序列

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.

Constraints:

• 1 <= text1.length <= 1000
• 1 <= text2.length <= 1000
• The input strings consist of lowercase English characters only.

這道題讓求最長相同的子序列，注意是子序列，不是子串，所以字符並不需要相連，但是字符順序還是需要保持的。LeetCode 之前也有題目需要借助求 LCS 來解題，比如 Delete Operation for Two Strings，當時博主還疑惑怎么 LeetCode 中沒有專門求 LCS 的題呢，這不，終於補上了。解題思路和上面那道是一模一樣，若搞懂了那道題，這道也就是沒什么難度了，這里是用動態規划 Dynamic Programing 來做，使用一個二維數組 dp，其中 dp[i][j] 表示 text1 的前i個字符和 text2 的前j個字符的最長相同的子序列的字符個數，這里大小初始化為 (m+1)x(n+1)，這里的m和n分別是 text1 和 text2 的長度。接下來就要找狀態轉移方程了，如何來更新 dp[i][j]，若二者對應位置的字符相同，表示當前的 LCS 又增加了一位，所以可以用 dp[i-1][j-1] + 1 來更新 dp[i][j]。否則若對應位置的字符不相同，由於是子序列，還可以錯位比較，可以分別從 text1 或者 text2 去掉一個當前字符，那么其 dp 值就是 dp[i-1][j] 和 dp[i][j-1]，取二者中的較大值來更新 dp[i][j] 即可，最終的結果保存在了 dp[m][n] 中，參見代碼如下：

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

討論：這道題跟之前那道 Longest Palindromic Subsequence 的解題思路幾乎是一模一樣，你品，你細品，一定要通過現象看本質，才能舉一反三，觸類旁通。

Github 同步地址:

https://github.com/grandyang/leetcode/issues/1143

類似題目：

Longest Palindromic Subsequence

Delete Operation for Two Strings

Shortest Common Supersequence

參考資料：

https://leetcode.com/problems/longest-common-subsequence/

https://leetcode.com/problems/longest-common-subsequence/discuss/348884/C%2B%2B-with-picture-O(nm)

https://leetcode.com/problems/longest-common-subsequence/discuss/351689/JavaPython-3-Two-DP-codes-of-O(mn)-and-O(min(m-n))-spaces-w-picture-and-analysis

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