> 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/interval-dp.md).

# 区间 DP

### 思想

区间 DP 以**区间长度**为阶段，`dp[i][j]` 表示区间 `[i, j]` 的最优解，从小区间推向大区间。

### 模板

```cpp
// dp[i][j]：合并区间 [i, j] 的最小代价
for (int len = 2; len <= n; len++)          // 区间长度
    for (int i = 1; i + len - 1 <= n; i++) {
        int j = i + len - 1;
        dp[i][j] = INF;
        for (int k = i; k < j; k++)         // 枚举分割点
            dp[i][j] = min(dp[i][j], dp[i][k] + dp[k+1][j] + cost(i, j));
    }
```

### 经典题：石子合并

每次合并相邻两堆，代价为两堆之和，求合并成一堆的最小总代价：

```cpp
dp[i][j] = min(dp[i][j], dp[i][k] + dp[k+1][j] + sum[j] - sum[i-1]);
```

### 复杂度

状态 O(n²)，转移 O(n)，共 **O(n³)**，适合 n ≤ 300。

### 要点

* 必须按**区间长度**从小到大枚举。
* 环形的石子合并可把数组复制一倍转为链。


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation by asking a question.

Perform an HTTP GET request on the following URL with the `ask` and `goal` query parameters:

```
GET https://docs.1l.lol/advanced-dp/interval-dp.md?ask=<question>&goal=<user_goal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is what the user is ultimately trying to achieve, the reason they need the answer. Sharing it helps GitBook give you a better, more relevant answer. A goal is most helpful when it describes the outcome the user wants rather than restating the question. For example, with `ask=how do I create an API token`, a goal like `automate deployments from our CI pipeline` lets GitBook tailor the answer to that use case.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.