Commit 8609925c authored by 24OI-bot's avatar 24OI-bot
Browse files

style: format markdown files with remark-lint

parent 0b9b8641

docs/graph/mdst.md

+110 −115
Original line number Diff line number Diff line 


## 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。
@@ -90,7 +89,6 @@ Tarjan的算法分为**收缩** 与 **伸展**两个过程。接下来先介绍* 
### 代码



```cpp



#include <bits/stdc++.h>



using namespace std;
@@ -124,14 +122,14 @@ struct Heap { 
Heap *merge(Heap *x, Heap *y) {

  if (!x) return y;

  if (!y) return x;

    if (x->e->w + x->constant > y->e->w + y->constant)

        swap(x, y);

  if (x->e->w + x->constant > y->e->w + y->constant) swap(x, y);

  x->push();

  x->rch = merge(x->rch, y);

    if (!x->lch || x->lch->rk < x->rch->rk)

        swap(x->lch, x->rch);

	if (x->rch) x->rk = x->rch->rk + 1;

	else x->rk = 1;

  if (!x->lch || x->lch->rk < x->rch->rk) swap(x->lch, x->rch);

  if (x->rch)

    x->rk = x->rch->rk + 1;

  else

    x->rk = 1;

  return x;

}

Edge *extract(Heap *&x) {
@@ -152,11 +150,12 @@ void contract() { 
  // 将图上的每一个结点与其相连的那些结点进行记录。

  for (int i = 1; i <= n; i++) {

    queue<Heap *> q;

        for (int j = 0; j < in[i].size(); j++) 

			    q.push(new Heap(&in[i][j]));

    for (int j = 0; j < in[i].size(); j++) q.push(new Heap(&in[i][j]));

    while (q.size() > 1) {

            Heap *u = q.front(); q.pop();

            Heap *v = q.front(); q.pop();

      Heap *u = q.front();

      q.pop();

      Heap *v = q.front();

      q.pop();

      q.push(merge(u, v));

    }

    Q[i] = q.front();
@@ -184,9 +183,9 @@ void contract() { 
ll expand(int x, int r);

ll expand_iter(int x) {

  ll r = 0;

    for (int u = nxt[x]; u != x; u = nxt[u])

	{

        if (ed[u]->w0 >= INF) return INF;

  for (int u = nxt[x]; u != x; u = nxt[u]) {

    if (ed[u]->w0 >= INF)

      return INF;

    else

      r += expand(ed[u]->v, u) + ed[u]->w0;

  }
@@ -194,40 +193,36 @@ ll expand_iter(int x) { 
}

ll expand(int x, int t) {

  ll r = 0;

    for (; x != t; x = fa[x])

	{

  for (; x != t; x = fa[x]) {

    r += expand_iter(x);

    if (r >= INF) return INF;

  }

  return r;

}

void link(int u, int v, int w) {

	in[v].push_back({u, v, w, w}); 

}

void link(int u, int v, int w) { in[v].push_back({u, v, w, w}); }



int main() {

  int rt;

  scanf("%d %d %d", &n, &m, &rt);

    for (int i = 0; i < m; i++) 

	{

  for (int i = 0; i < m; i++) {

    int u, v, w;

    scanf("%d %d %d", &u, &v, &w);

    link(u, v, w);

  }

  // 保证强连通

    for (int i = 1; i <= n; i++) 

        link(i > 1 ? i - 1 : n, i, INF);

  for (int i = 1; i <= n; i++) link(i > 1 ? i - 1 : n, i, INF);

  contract();

  ll ans = expand(rt, n);

    if (ans >= INF) puts("-1");

    else printf("%lld\n", ans);                                                       

  if (ans >= INF)

    puts("-1");

  else

    printf("%lld\n", ans);

  return 0;

}

```





## 参考文献



Uri Zwick. (2013), [Directed Minimum Spanning Trees](http://www.cs.tau.ac.il/~zwick/grad-algo-13/directed-mst.pdf) , Lecture notes on“Analysis of Algorithms”



https://riteme.site/blog/2018-6-18/mdst.html#_3
 
 <https://riteme.site/blog/2018-6-18/mdst.html#_3>