> 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/algorithm-techniques/heap.md).

# 堆与优先队列

### 堆的概念

堆是一棵**完全二叉树**：

* **小根堆**：每个结点 ≤ 其孩子（根最小）
* **大根堆**：每个结点 ≥ 其孩子（根最大）

插入 / 删除堆顶 O(log n)，取堆顶 O(1)。

### C++ priority\_queue

```cpp
#include <queue>

priority_queue<int> pq;                                // 大根堆（默认）
priority_queue<int, vector<int>, greater<int>> pq2;    // 小根堆

pq.push(3); pq.push(1); pq.push(5);
cout << pq.top();   // 5
pq.pop();
```

### 用 pair 自动排序

```cpp
priority_queue<pair<int,int>> pq;   // 先比 first，再比 second
pq.push({2, 5});
pq.push({2, 3});
cout << pq.top().second;   // 3
```

### 手写堆（数组）

下标从 1 开始，结点 i 的孩子为 2i、2i+1，用上浮 / 下沉维护。

### 应用

求第 k 大/小、动态最值、哈夫曼树、Dijkstra、合并有序序列。


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