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最小直径生成树

最小树形图

有向图上的最小生成树（Minimum Directed Spanning Tree）称为最小树形图。

常用的算法是朱刘算法（也称 Edmonds 算法），可以在 时间内解决最小树形图问题。

流程

1. 对于每个点，选择它入度最小的那条边
2. 如果没有环，算法终止；否则进行缩环并更新其他点到环的距离。

代码

bool solve() {
  ans = 0;
  int u, v, root = 0;
  for (;;) {
    f(i, 0, n) in[i] = 1e100;
    f(i, 0, m) {
      u = e[i].s;
      v = e[i].t;
      if (u != v && e[i].w < in[v]) {
        in[v] = e[i].w;
        pre[v] = u;
      }
    }
    f(i, 0, m) if (i != root && in[i] > 1e50) return 0;
    int tn = 0;
    memset(id, -1, sizeof id);
    memset(vis, -1, sizeof vis);
    in[root] = 0;
    f(i, 0, n) {
      ans += in[i];
      v = i;
      while (vis[v] != i && id[v] == -1 && v != root) {
        vis[v] = i;
        v = pre[v];
      }
      if (v != root && id[v] == -1) {
        for (int u = pre[v]; u != v; u = pre[u]) id[u] = tn;
        id[v] = tn++;
      }
    }
    if (tn == 0) break;
    f(i, 0, n) if (id[i] == -1) id[i] = tn++;
    f(i, 0, m) {
      u = e[i].s;
      v = e[i].t;
      e[i].s = id[u];
      e[i].t = id[v];
      if (e[i].s != e[i].t) e[i].w -= in[v];
    }
    n = tn;
    root = id[root];
  }
  return ans;
}

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