> 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/graph-theory/union-find.md).

# 并查集

### 并查集的作用

维护若干个**不相交集合**，支持快速**合并**与**查询所属集合**。

### 基本实现

```cpp
int fa[N];

void init(int n) { for (int i = 1; i <= n; i++) fa[i] = i; }

int find(int x) {              // 路径压缩
    return fa[x] == x ? x : fa[x] = find(fa[x]);
}

void unite(int a, int b) {     // 合并
    fa[find(a)] = find(b);
}
```

### 路径压缩

`find` 时把路径上的点直接连到根，后续查询接近 O(1)。

### 按秩合并（可选）

每次把小树接到大树下，避免退化：

```cpp
void unite(int a, int b) {
    int ra = find(a), rb = find(b);
    if (ra == rb) return;
    if (sz[ra] < sz[rb]) swap(ra, rb);
    fa[rb] = ra;
    sz[ra] += sz[rb];
}
```

两者结合，单次操作均摊接近 **O(α(n))**（反阿克曼函数，极小）。

### 应用

* 连通性问题、判断是否在同一集合
* Kruskal 最小生成树
* 判断无向图环


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