> 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/topological-sort.md).

# 拓扑排序与 DAG

### 有向无环图（DAG）

DAG 是没有环的有向图，很多“先后顺序”问题都能建成 DAG。

### 拓扑排序

把顶点排成线性序列，使每条边 `u → v` 中 u 都在 v 之前。

#### Kahn 算法（BFS，入度法）

```cpp
vector<int> adj[N];
int indeg[N], order[N], idx = 0;

void topo(int n) {
    queue<int> q;
    for (int i = 1; i <= n; i++)
        if (indeg[i] == 0) q.push(i);
    while (!q.empty()) {
        int u = q.front(); q.pop();
        order[idx++] = u;
        for (int v : adj[u])
            if (--indeg[v] == 0) q.push(v);
    }
    if (idx < n) cout << "存在环";   // 未输出全部，说明有环
}
```

#### DFS 法

对每个点 DFS，回溯时把点加入序列，最后翻转（需另检测环）。

### 应用

* 任务调度、课程安排（先修关系）
* 有向图判环
* 在 DAG 上做动态规划（如最长路）

### DAG 上的 DP

```cpp
dp[v] = max(dp[v], dp[u] + w);   // 按拓扑序递推
```


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