HDU-3416 Marriage Match IV(最短路，最大流)

题目链接：HDU-3416 Marriage Match IV

题意

给出一个有向图$G$，起点$s$和终点$t$，问$s$到$t$有多少条完全不同的最短路径(即长度相同，没有公共边)。

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思路

求最短路的条数可以用最大流，不过要去掉原图中不可能属于最短路的边。

先用dijkstra求出各个点$u$与起点$s$的最短距离$dis[u]$，若一条边$(u,v,w)$满足$dis[v]-dis[u]==w$，说明这条边是$s\to v$最短路上的边，将这样的边加入流网络，容量为1，最后求出以起点$s$为源点终点$t$为汇点的最大流就是答案。

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代码实现

#include <iostream>
#include <cstdio>
#include <cstring>
#include <queue>
#include <utility>
#include <vector>
using std::pair;
using std::queue;
const int INF = 0x3f3f3f3f, N = 1100, M = 220000;
int head[N], head2[N], d[N];
int s, t, tot, tot2, maxflow;
typedef pair<int, int> P;
int dis[N];
struct Edge
{
    int to, cap, nex;
} edge[M], edge2[M];
queue<int> q;
void add2(int x, int y, int z) {
    edge2[++tot2].to = y, edge2[tot2].cap = z, edge2[tot2].nex = head2[x], head2[x] = tot2;
}
void dijkstra() {
    std::priority_queue<P, std::vector<P>, std::greater<P> > que;
    memset(dis, INF, sizeof(dis));
    dis[s] = 0;
    que.push(P(0, s));
    while (!que.empty()) {
        P p = que.top(); que.pop();
        int v = p.second;
        if (p.first > dis[v]) continue;
        for (int i = head2[v], u; i; i = edge2[i].nex) {
            u = edge2[i].to;
            if (dis[v] + edge2[i].cap < dis[u]) {
                dis[u] = dis[v] + edge2[i].cap;
                que.push(P(dis[u], u));
            }
        }
    }
}
void add(int x, int y, int z) {
    edge[++tot].to = y, edge[tot].cap = z, edge[tot].nex = head[x], head[x] = tot;
    edge[++tot].to = x, edge[tot].cap = 0, edge[tot].nex = head[y], head[y] = tot;
}
bool bfs() {
    memset(d, 0, sizeof(d));
    while (q.size()) q.pop();
    q.push(s); d[s] = 1;
    while (q.size()) {
        int x = q.front(); q.pop();
        for (int i = head[x]; i; i = edge[i].nex) {
            int v = edge[i].to;
            if (edge[i].cap && !d[v]) {
                q.push(v);
                d[v] = d[x] + 1;
                if (v == t) return true;
            }
        }
    }
    return false;
}
int dinic(int x, int flow) {
    if (x == t) return flow;
    int rest = flow, k;
    for (int i = head[x]; i && rest; i = edge[i].nex) {
        int v = edge[i].to;
        if (edge[i].cap && d[v] == d[x] + 1) {
            k = dinic(v, std::min(rest, edge[i].cap));
            if (!k) d[v] = 0;
            edge[i].cap -= k;
            edge[i^1].cap += k;
            rest -= k;
        }
    }
    return flow - rest;
}
void init() {
    tot = 1, tot2 = 0, maxflow = 0;
    memset(head, 0, sizeof(head));
    memset(head2, 0, sizeof(head2));
}

int main() {
    int T;
    scanf("%d", &T);
    while (T--) {
        int n, m;
        scanf("%d %d", &n, &m);
        init();
        for (int i = 0, u, v, z; i < m; i++) {
            scanf("%d %d %d", &u, &v, &z);
            if (u == v) continue;
            add2(u, v, z);
        }
        scanf("%d %d", &s, &t);
        dijkstra();
        for (int i = 1; i <= n; i++) {
            for (int j = head2[i], u; j; j = edge2[j].nex) {
                u = edge2[j].to;
                if (dis[u] - dis[i] == edge2[j].cap) add(i, u, 1);
            }
        }
        while (bfs()) maxflow += dinic(s, INF);
        printf("%d\n", maxflow);
    }
    return 0;
}