> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/advanced-dp/lcs.md).

# 最长公共子序列（LCS）

### 问题

求两个序列的最长公共子序列长度。

### DP O(nm)

```cpp
for (int i = 1; i <= n; i++)
    for (int j = 1; j <= m; j++) {
        if (a[i] == b[j]) dp[i][j] = dp[i-1][j-1] + 1;
        else dp[i][j] = max(dp[i-1][j], dp[i][j-1]);
    }
cout << dp[n][m];
```

### 转移含义

* 若 a\[i] == b\[j]：可以接在 dp\[i-1]\[j-1] 后。
* 否则：舍弃 a 或 b 的最后一个，取较优。

### 输出具体序列

从 dp\[n]\[m] 回溯：

```cpp
string s;
int i = n, j = m;
while (i > 0 && j > 0) {
    if (a[i] == b[j]) { s += a[i]; i--; j--; }
    else if (dp[i-1][j] >= dp[i][j-1]) i--;
    else j--;
}
reverse(s.begin(), s.end());
```

### 对比

* LIS：一个序列，O(n log n)
* LCS：两个序列，O(nm)，空间可用滚动数组优化到 O(m)


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