> 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/algorithms/dynamic-programming.md).

# 动态规划入门（一维 DP）

### 核心思想

动态规划（DP）通过把问题拆成**重叠子问题**，保存子问题的答案（**记忆化**），避免重复计算，从而高效求解。

### 使用条件

* **最优子结构**：整体最优由子问题最优构成
* **重叠子问题**：子问题被重复使用
* **无后效性**：当前状态确定后，之后的决策与如何到达该状态无关

### 解题步骤

1. 定义**状态**（dp\[i] 表示什么）
2. 确定**初始值**
3. 推导**状态转移方程**
4. 确定**遍历顺序**与答案

### 例子：数字三角形（最大路径和）

```cpp
// dp[i][j] = a[i][j] + max(dp[i+1][j], dp[i+1][j+1])
for (int j = 1; j <= n; j++) dp[n][j] = a[n][j];
for (int i = n - 1; i >= 1; i--)
    for (int j = 1; j <= i; j++)
        dp[i][j] = a[i][j] + max(dp[i + 1][j], dp[i + 1][j + 1]);
cout << dp[1][1];
```

### 例子：最长不下降子序列长度（LIS 基础版）

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

### 与递推的关系

递推是从已知推未知，DP 是把递推用于求最值/方案数，并强调状态定义与转移。


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