uva 1349 Optimal Bus Route Design(拆点,费用流)

题目链接

题意：给出(n(1 leq n leq 100))个点，每个点与其他一些点相连，边上有权值，现在需要选一些边将所有点包括在一个环内(有且仅有一个环包含每个点，可以有多个环)，使得所选边权值之和最小。

题解：对于每个点存在唯一环包括他，等价于每个点存在唯一后继，对于唯一性，可以想到二分图，将原来每个点拆为两个点，然后加个源和汇s，t，在将边权转化为费用跑费用流就行了。

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cstring>
#include<vector>
#include<queue>
#include<stack>
using namespace std;
#define rep(i,a,n) for (int i=a;i<n;i++)
#define per(i,a,n) for (int i=n-1;i>=a;i--)
#define pb push_back
#define fi first
#define se second
typedef vector<int> VI;
typedef long long ll;
typedef pair<int,int> PII;
const int inf=0x3fffffff;
const ll mod=1000000007;
const int MAXN =200+10;
const int MAXM = 21000;
const int INF = 1e9;
struct Edge
{
    int to,next,cap,flow,cost;
}edge[MAXM];
int head[MAXN],tol;
int pre[MAXN],dis[MAXN];
bool vis[MAXN];
int N;//节点总个数，节点编号从0~N-1

void addedge(int u,int v,int cap,int cost)
{
    edge[tol].to = v;edge[tol].cap = cap;edge[tol].cost = cost;edge[tol].flow = 0;
    edge[tol].next = head[u];head[u] = tol++;
    edge[tol].to = u;edge[tol].cap = 0;edge[tol].cost = -cost;edge[tol].flow = 0;
    edge[tol].next = head[v];head[v] = tol++;
}


bool spfa(int s,int t)
{
    queue<int> q;
    for(int i = 0;i < N;i++)
    {
        dis[i] = INF;
        vis[i] = false;
        pre[i] = -1;
    }
    dis[s] = 0;
    vis[s] = true;
    q.push(s);
    while(!q.empty())
    {
        int u = q.front();
        q.pop();
        vis[u] = false;
        for(int i = head[u]; i != -1;i = edge[i].next)
        {
            int v = edge[i].to;
            if(edge[i].cap > edge[i].flow &&
               dis[v] > dis[u] + edge[i].cost )
            {
                dis[v] = dis[u] + edge[i].cost;
                pre[v] = i;
                if(!vis[v])
                {
                    vis[v] = true;
                    q.push(v);
                }
            }
        }
    }
    if(pre[t] == -1)return false;
    else return true;
}
//返回的是最大流，cost存的是最小费用
int minCostMaxflow(int s,int t,int &cost)
{
    int flow = 0;
    cost = 0;
    while(spfa(s,t))
    {
        int Min = INF;
        for(int i = pre[t];i != -1;i = pre[edge[i^1].to])
        {
            if(Min > edge[i].cap - edge[i].flow)
                Min = edge[i].cap - edge[i].flow;
        }
        for(int i = pre[t];i != -1;i = pre[edge[i^1].to])
        {
            edge[i].flow += Min;
            edge[i^1].flow -= Min;
            cost += edge[i].cost * Min;
        }
        flow += Min;
    }
    return flow;
}


int main()
{
    int n;
    while(~scanf("%d",&n)&&n)
    {
        int st=0,ed=2*n+1;
        N=ed+1;
        tol=0;
        rep(i,0,ed+1) head[i]=-1;
        rep(i,1,n+1)
        {
            int p;
            while(~scanf("%d",&p))
            {
                if(!p) break;
                int w;
                scanf("%d",&w);
                addedge(i,n+p,1,w);
            }
        }
        rep(i,1,n+1) addedge(st,i,1,0),addedge(n+i,ed,1,0);
        int cost=0;
        int t=minCostMaxflow(st,ed,cost);
        if(t<n) puts("N");
        else printf("%d
",cost);
    }
    return 0;
}