> 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/divide-and-conquer.md).

# 分治算法

### 思想

分治（分而治之）把大问题**分解**成若干个规模更小的同类子问题，**递归求解**后再**合并**结果。

三步：**分解 → 解决 → 合并**。

### 经典应用

* **归并排序**：二分后合并两个有序段
* **快速排序**：以基准分区后递归
* **二分查找**：每次排除一半
* 求最大子段和、大整数乘法等

### 归并与快排的对比

|     | 归并排序       | 快速排序  |
| --- | ---------- | ----- |
| 分法  | 从中间等分      | 按基准值分 |
| 合并  | 需要合并       | 无需合并  |
| 稳定性 | 稳定         | 不稳定   |
| 最坏  | O(n log n) | O(n²) |

### 复杂度分析

若每次把问题分成规模 n/2 的两个子问题，合并代价 O(n)，则 `T(n) = 2T(n/2) + O(n)`，解得 **O(n log n)**。

### 分治与递归

分治通常用递归实现，但分治是一种**算法思想**，递归只是实现手段。


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