HUD3416 Marriage Match IV

胡扯

看到这道题写最短路也写网络流，正好拿过来写写

难度很低，大佬轻喷/kk

思路

最短路可行边+网络最大流

手残写错了几个地方调了半个上午

题意就是找题目给定起点终点之间最短路的条数。

有一个很显然的性质，设起点为 \(s\), 终点为 \(t\)， \(dis_{u,v}\) 表示 \(u\) 到 \(v\) 的最短路，\(val_{x,y}\) 表示边 \((x,y)\) 的长度，则对于一条可以在最短路上的边\((x,y)\)，一定有 \(dis_{s,x}+val_{x,y}+dis_{y,t}=dis_{s,t}\)。

有了这个性质，我们就可以将最短路上的边重新建图，边的流量为 \(1\)，然后在图上跑最大流，得到的最大流就是最短路的条数。

求最短路可以用 \(Dij\) 跑出 \(s\) 到每个点的最短路，同时建反向边，跑出 \(t\) 到每个点的最短路（实际上是每个点到 \(t\) 的最短路），再进行上述判断即可。

重边不用判，因为说不定这条重边也可以在最短路上。

代码

注释不删了，好让自己明白自己有多菜

#include <cmath>
#include <queue>
#include <cstdio>
#include <vector>
#include <cstring>
#include <iostream>
#include <algorithm>
using namespace std;

const int A = 1e5 + 11;
const int B = 2e3 + 11;
const int mod = 1e9 + 7;
const int inf = 0x3f3f3f3f;

inline int read() {
  char c = getchar();
  int x = 0, f = 1;
  for ( ; !isdigit(c); c = getchar()) if (c == '-') f = -1;
  for ( ; isdigit(c); c = getchar()) x = x * 10 + (c ^ 48);
  return x * f;
}

struct edge { int to, nxt, val; } e[A << 1];
int n, m, cnt, flag[B][B], endd, head[2][A], dis[2][A], vis[A];

inline void add(int tg, int from, int to, int val) {
  e[++cnt].to = to;
  e[cnt].val = val;
  e[cnt].nxt = head[tg][from];
  head[tg][from] = cnt;
}

struct node {
  int x, y;
  bool operator < (const node &b) const {
    return y > b.y;
  }
};

inline void Dij(int s, int tg) {
  priority_queue <node> Q;
  while (!Q.empty()) Q.pop();
  memset(vis, 0, sizeof(vis));
  memset(dis[tg], inf, sizeof(dis[tg]));
  dis[tg][s] = 0;
  Q.push((node){s, 0});
  while (!Q.empty()) {
    int now = Q.top().x; Q.pop();
    if (vis[now]) continue;
    vis[now] = 1;
    for (int i = head[tg][now]; i; i = e[i].nxt) {
      int to = e[i].to;
      if (dis[tg][now] + e[i].val < dis[tg][to]) {
        dis[tg][to] = dis[tg][now] + e[i].val;
        Q.push((node){to, dis[tg][to]});
      }
    }
  }
//  for (int i = 1; i <= n; i++) cout << dis[tg][i] << " ";
//  puts("");
}

struct Max_flow {
  struct edge { int to, nxt, val; } edd[A << 1];
  int ccnt = 1, cur[A], inq[A], dep[A], hdd[A], vis[A];
  
  inline void init() {
    ccnt = 0;
    memset(hdd, 0, sizeof(hdd));
    memset(cur, 0, sizeof(cur));
  }
  
  inline void add(int from, int to, int val) {
    edd[++ccnt].to = to;
    edd[ccnt].val = val;
    edd[ccnt].nxt = hdd[from];
    hdd[from] = ccnt;
  }
  
  inline bool bfs(int s, int t) {
    queue <int> Q;
    for (int i = 1; i <= n; i++) 
      cur[i] = hdd[i], dep[i] = inf, inq[i] = 0;
    Q.push(s), dep[s] = 0, inq[s] = 1;
    while (!Q.empty()) {
      int x = Q.front(); Q.pop(), inq[x] = 0;
      for (int i = hdd[x]; i; i = edd[i].nxt) {
        int to = edd[i].to;
        if (dep[x] + 1 < dep[to] && edd[i].val) {
          dep[to] = dep[x] + 1;
          if (!inq[to]) Q.push(to), inq[to] = 1;
        }
      }
    }
    return dep[t] != inf;
  }
  
  int dfs(int x, int flow) {
    if (x == endd) return flow;
    for (int i = cur[x], tmp = 0; i; i = edd[i].nxt) {
      cur[x] = i;
      int to = edd[i].to;
      if (dep[to] == dep[x] + 1 && edd[i].val) {
        if (tmp = dfs(to, min(edd[i].val, flow))) {
          edd[i].val -= tmp, edd[i ^ 1].val += tmp;
          return tmp;
        }
      }
    }
    return 0;
  }
} Flow;

inline void solve() {
  memset(flag, 0, sizeof(flag));
  memset(head, 0, sizeof(head));
  Flow.init();
  cnt = 0;
  n = read(), m = read();
  int tot = 0;
  for (int i = 1; i <= m; i++) {
    int u = read(), v = read(), w = read();
    add(0, u, v, w);
    add(1, v, u, w);
  }
  int s = read(), t = read();
  Dij(s, 0), Dij(t, 1);
//  for (int i = 1; i <= n; i++) {
//    for (int j = 1; j <= n; j++) {
//      cout << flag[i][j] << " ";
//    }
//    puts("");
//  }
  for (int i = 1; i <= n; i++) {
    for (int j = head[0][i]; j; j = e[j].nxt) {
//      cout << "ij " << i << " " << e[j].to << " ";
//      cout << dis[0][i] << " " << e[j].val << " " << dis[1][e[j].to] << " " << dis[0][t] << '\n';
      if (dis[0][i] + e[j].val + dis[1][e[j].to] == dis[0][t]) {
        Flow.add(i, e[j].to, 1);
        Flow.add(e[j].to, i, 0);
      }
    }
  }
  endd = t;
  int ans = 0, now = 0;
//  cout << Flow.ccnt << '\n';
  while (Flow.bfs(s, t)) while (now = Flow.dfs(s, inf)) ans += now;
  cout << ans << '\n';
  return;
}

signed main() {
  int T = read();
  while (T--) solve();
}