> 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/mst.md).

# 最小生成树

### 概念

在连通带权无向图中，选出一棵包含所有顶点、边权和最小的树，即**最小生成树（MST）**。

### Kruskal（并查集 + 排序）

按边权从小到大选边，若两端不在同一集合则加入。复杂度 **O(m log m)**。

```cpp
struct Edge { int u, v, w; };
bool cmp(Edge a, Edge b) { return a.w < b.w; }

int find(int x) { return fa[x] == x ? x : fa[x] = find(fa[x]); }

int kruskal() {
    sort(e, e + m, cmp);
    for (int i = 1; i <= n; i++) fa[i] = i;
    int total = 0, cnt = 0;
    for (int i = 0; i < m && cnt < n - 1; i++) {
        int ru = find(e[i].u), rv = find(e[i].v);
        if (ru != rv) { fa[ru] = rv; total += e[i].w; cnt++; }
    }
    return cnt == n - 1 ? total : -1;   // -1 表示不连通
}
```

### Prim（邻接矩阵 / 堆优化）

从任一点出发，每次加入距离已选集合最近的顶点。复杂度 O(n²) 或 O(m log n)。

### 对比

|     | Kruskal    | Prim             |
| --- | ---------- | ---------------- |
| 依赖  | 并查集        | 起点扩展             |
| 适合  | 稀疏图        | 稠密图              |
| 复杂度 | O(m log m) | O(n²)/O(m log n) |


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