最小费用流: uva 1658

最新推荐文章于 2018-09-06 11:59:59 发布

转载最新推荐文章于 2018-09-06 11:59:59 发布 · 263 阅读

本文详细介绍了最小费用流算法的实现细节，包括如何通过Bellman-Ford算法寻找增广路径来求解特定流量条件下的最小费用流问题。此外，还提供了完整的C++代码示例，帮助读者更好地理解算法原理。

最小费用流可以求

某个流量的最小费用流，

也可以直接求最大流量的最小费用流

某个流量的最小费用流：

 int MincostFlow(int s, int t, int flow_limit, int& cost) {
        int flow = 0; cost = 0;
        while(flow < flow_limit && BellmanFord(s, t, flow_limit, flow, cost));
        return flow;
    }

MCMF

int minflow(int s,int t,long long &cost){
    int flow=0;cost=0;
    while(BellmanFord(s,t,flow,cost));
    return flow
}

套上模板。。

#include<iostream>
#include<string>
#include<cstring>
#include<algorithm>
#include<cstdio>
#include<vector>
#include<queue>
#include<bits/stdc++.h>
using namespace std;
#define mem(a,b) memset(a,b,sizeof(a));
#define sf scanf
#define pf printf
#define LL long long
const int maxn = 2000 + 10;
const int INF = 1000000000;
struct Edge {
  int from, to, cap, flow, cost;
  Edge(int u, int v, int c, int f, int w):from(u),to(v),cap(c),flow(f),cost(w) {}
};
struct MCMF {
    int n, m;
    vector<Edge> edges;
    vector<int> G[maxn];
    int inq[maxn];         // 是否在队列中
    int d[maxn];           // Bellman-Ford
    int p[maxn];           // 上一条弧
    int a[maxn];           // 可改进量
    void init(int n) {
        this->n = n;
        for(int i = 0; i < n; i++) G[i].clear();
        edges.clear();
    }
    void addedge(int from, int to, int cap, int cost) {
        edges.push_back(Edge(from, to, cap, 0, cost));
        edges.push_back(Edge(to, from, 0, 0, -cost));
        m = edges.size();
        G[from].push_back(m-2);
        G[to].push_back(m-1);
    }
    bool BellmanFord(int s, int t, int flow_limit, int& flow, int& cost) {
        for(int i = 0; i < n; i++) d[i] = INF;
        memset(inq, 0, sizeof(inq));
        d[s] = 0; inq[s] = 1; p[s] = 0; a[s] = INF;
        queue<int> Q;
        Q.push(s);
        while(!Q.empty()) {
          int u = Q.front(); Q.pop();
          inq[u] = 0;
          for(int i = 0; i < G[u].size(); i++) {
            Edge& e = edges[G[u][i]];
            if(e.cap > e.flow && d[e.to] > d[u] + e.cost) {
                d[e.to] = d[u] + e.cost;
                p[e.to] = G[u][i];
                a[e.to] = min(a[u], e.cap - e.flow);
                if(!inq[e.to]) { Q.push(e.to); inq[e.to] = 1; }
            }
          }
        }
        if(d[t] == INF) return false;
        if(flow + a[t] > flow_limit) a[t] = flow_limit - flow;
        flow += a[t];
        cost += d[t] * a[t];
        for(int u = t; u != s; u = edges[p[u]].from) {
          edges[p[u]].flow += a[t];
          edges[p[u]^1].flow -= a[t];
        }
        return true;
    }
    // 需要保证初始网络中没有负权圈
    int MincostFlow(int s, int t, int flow_limit, int& cost) {
        int flow = 0; cost = 0;
        while(flow < flow_limit && BellmanFord(s, t, flow_limit, flow, cost));
        return flow;//返回的流量，但其实是没有用的，求的最小费用在&cost中求了
    }
};
MCMF g;
int main(){
    int n,m;
    while(~sf("%d%d",&n,&m)){
       g.init(n*2-2);

        for(int i=2;i<=n-1;++i){
            g.addedge(i-1,i+n-2,1,0);
        }
        // 点2~n-1拆成弧i->i'，前者编号为1~n-2，后者编号为n~2n-2
        for(int i=1;i<=m;++i){
            int u,v,c;
            sf("%d%d%d",&u,&v,&c);
            if(u!=1&&u!=n)u+=n-2;
            else u--;
            v--;
            g.addedge(u,v,1,c);
        }
        int cost;
        g.MincostFlow(0,n-1,2,cost);//s为0，  t为n-1
        pf("%d\n",cost);

    }

}