> 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/hash-table.md).

# 哈希表

### 思想

哈希表通过**哈希函数**把键映射到数组下标，实现平均 O(1) 的插入与查找。冲突用拉链或开放寻址解决。

### C++ 哈希容器

```cpp
#include <unordered_map>
#include <unordered_set>

unordered_map<string, int> mp;   // 平均 O(1)
mp["apple"] = 3;
if (mp.count("apple")) cout << mp["apple"];

unordered_set<int> s;            // 去重集合
s.insert(5);
if (s.count(5)) cout << "存在";
```

### map vs unordered\_map

|     | map      | unordered\_map  |
| --- | -------- | --------------- |
| 底层  | 红黑树      | 哈希表             |
| 有序  | 是        | 否               |
| 复杂度 | O(log n) | 平均 O(1)，最坏 O(n) |

### 手写哈希（取模法）

```cpp
const int MOD = 1000003;
int head[MOD], nxt[N], val[N], tot;

int hashKey(int x) { return (x % MOD + MOD) % MOD; }
```

### 应用

判重、统计出现次数、字符串哈希、构建映射。

### 注意

出题人可能构造数据让 unordered\_map 退化，可加随机种子或改用 map。


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