HDU 4067 Random Maze 花消流
HDU 4067 Random Maze 费用流
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Total Submission(s): 1089 Accepted Submission(s): 379
题意：给出一个有n个点m条边的有向图，现在要从图中删去一些边使得图满足以下条件：
1.图中只有一个入口(s)和一个出口(t)
2.所以边都是单向的(这个原图已经保证了)
3.对入口(s)而言，其出度=入度+1
4.对出口(t)而言，其入度=出度+1
5.对入口(s)、出口(t)以外的点，其入度=出度
而对图中每条边，保留这条边或者删去这条边都有相应的花费，分别为a、b，求使得该图满足条件的最小花费。
解法：
重新为每个边建立权值！
1.如果有边（u,v,a,b) 并且a<=b，那么建立边（v,u,1，b-a) sum+=a,如果要恢复此边需花费 b-a 的费用；in[u]++,out[v]++;
2.如果有边（u,v,a,b) 并且a>b，那么建立边（u,v,1，a-b) sum+=b,如果要删除此边需花费 a-b 的费用；
3.另需t和s 建立一条虚边，in[s]++,out[t]++;
最后建立一个超级源点S 和超级汇点T,则如果对于点 i in[i]>out[i] 那么建立边（S,i,in[i]-out[i]，0);
相反 建立边（i,T,out[i]-in[i]，0);
最小费用最大流后，如果max_flow=到汇点t所有容量只和 答案即为：sum+max_flow;
否则 impossible；
Random Maze
Time Limit: 10000/3000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)Total Submission(s): 1089 Accepted Submission(s): 379
Problem Description
In the game “A Chinese Ghost Story”, there are many random mazes which have some characteristic：
1.There is only one entrance and one exit.
2.All the road in the maze are unidirectional.
3.For the entrance, its out-degree = its in-degree + 1.
4.For the exit, its in-degree = its out-degree + 1.
5.For other node except entrance and exit, its out-degree = its in-degree.
There is an directed graph, your task is removing some edge so that it becomes a random maze. For every edge in the graph, there are two values a and b, if you remove the edge, you should cost b, otherwise cost a.
Now, give you the information of the graph, your task if tell me the minimum cost should pay to make it becomes a random maze.
1.There is only one entrance and one exit.
2.All the road in the maze are unidirectional.
3.For the entrance, its out-degree = its in-degree + 1.
4.For the exit, its in-degree = its out-degree + 1.
5.For other node except entrance and exit, its out-degree = its in-degree.
There is an directed graph, your task is removing some edge so that it becomes a random maze. For every edge in the graph, there are two values a and b, if you remove the edge, you should cost b, otherwise cost a.
Now, give you the information of the graph, your task if tell me the minimum cost should pay to make it becomes a random maze.
Input
The first line of the input file is a single integer T.
The rest of the test file contains T blocks.
For each test case, there is a line with four integers, n, m, s and t, means that there are n nodes and m edges, s is the entrance's index, and t is the exit's index. Then m lines follow, each line consists of four integers, u, v, a and b, means that there is an edge from u to v.
2<=n<=100, 1<=m<=2000, 1<=s, t<=n, s != t. 1<=u, v<=n. 1<=a, b<=100000
The rest of the test file contains T blocks.
For each test case, there is a line with four integers, n, m, s and t, means that there are n nodes and m edges, s is the entrance's index, and t is the exit's index. Then m lines follow, each line consists of four integers, u, v, a and b, means that there is an edge from u to v.
2<=n<=100, 1<=m<=2000, 1<=s, t<=n, s != t. 1<=u, v<=n. 1<=a, b<=100000
Output
For each case, if it is impossible to work out the random maze, just output the word “impossible”, otherwise output the minimum cost.(as shown in the sample output)
Sample Input
2 2 1 1 2 2 1 2 3 5 6 1 4 1 2 3 1 2 5 4 5 5 3 2 3 3 2 6 7 2 4 7 6 3 4 10 5
Sample Output
Case 1: impossible Case 2: 27
Source
The 36th ACM/ICPC Asia Regional
Fuzhou Site —— Online Contest
题意：给出一个有n个点m条边的有向图，现在要从图中删去一些边使得图满足以下条件：
1.图中只有一个入口(s)和一个出口(t)
2.所以边都是单向的(这个原图已经保证了)
3.对入口(s)而言，其出度=入度+1
4.对出口(t)而言，其入度=出度+1
5.对入口(s)、出口(t)以外的点，其入度=出度
而对图中每条边，保留这条边或者删去这条边都有相应的花费，分别为a、b，求使得该图满足条件的最小花费。
解法：
重新为每个边建立权值！
1.如果有边（u,v,a,b) 并且a<=b，那么建立边（v,u,1，b-a) sum+=a,如果要恢复此边需花费 b-a 的费用；in[u]++,out[v]++;
2.如果有边（u,v,a,b) 并且a>b，那么建立边（u,v,1，a-b) sum+=b,如果要删除此边需花费 a-b 的费用；
3.另需t和s 建立一条虚边，in[s]++,out[t]++;
最后建立一个超级源点S 和超级汇点T,则如果对于点 i in[i]>out[i] 那么建立边（S,i,in[i]-out[i]，0);
相反 建立边（i,T,out[i]-in[i]，0);
最小费用最大流后，如果max_flow=到汇点t所有容量只和 答案即为：sum+max_flow;
否则 impossible；
//546MS 512K
#include<iostream>
#include<algorithm>
#include<cstring>
#include<queue>
#include<cstdio>
using namespace std;
const int MAXN=8007;
const int inf=1<<29;
int pre[MAXN]; // pre[v] = k：在增广路上，到达点v的边的编号为k
int dis[MAXN]; // dis[u] = d：从起点s到点u的路径长为d
int vis[MAXN]; // inq[u]：点u是否在队列中
int path[MAXN];
int head[MAXN];
int NE,tot,ans,max_flow;
int in[107];
struct node
{
int u,v,cap,cost,next;
} Edge[MAXN<<2];
void addEdge(int u,int v,int cap,int cost)
{
Edge[NE].u=u;
Edge[NE].v=v;
Edge[NE].cap=cap;
Edge[NE].cost=cost;
Edge[NE].next=head[u];
head[u]=NE++;
Edge[NE].v=u;
Edge[NE].u=v;
Edge[NE].cap=0;
Edge[NE].cost=-cost;
Edge[NE].next=head[v];
head[v]=NE++;
}
int SPFA(int s,int t) // 源点为0，汇点为sink。
{
int i;
for(i=s;i<=t;i++) dis[i]=inf;
memset(vis,0,sizeof(vis));
memset(pre,-1,sizeof(pre));
dis[s] = 0;
queue<int>q;
q.push(s);
vis[s] =1;
while(!q.empty()) // 这里最好用队列，有广搜的意思，堆栈像深搜。
{
int u =q.front();
q.pop();
for(i=head[u]; i!=-1;i=Edge[i].next)
{
int v=Edge[i].v;
if(Edge[i].cap >0&& dis[v]>dis[u]+Edge[i].cost)
{
dis[v] = dis[u] + Edge[i].cost;
pre[v] = u;
path[v]=i;
if(!vis[v])
{
vis[v] =1;
q.push(v);
}
}
}
vis[u] =0;
}
if(pre[t]==-1)
return 0;
return 1;
}
void end(int s,int t)
{
int u, sum = inf;
for(u=t; u!=s; u=pre[u])
{
sum = min(sum,Edge[path[u]].cap);
}
max_flow+=sum; //记录最大流
for(u = t; u != s; u=pre[u])
{
Edge[path[u]].cap -= sum;
Edge[path[u]^1].cap += sum;
ans += sum*Edge[path[u]].cost; // cost记录的为单位流量费用，必须得乘以流量。
}
}
int main()
{
int n,m,s,t,tt,cas=1;
scanf("%d",&tt);
while(tt--)
{
memset(head,-1,sizeof(head));
memset(in,0,sizeof(in));
NE=ans=max_flow=0;
int sum=0,u,v,a,b;
scanf("%d%d%d%d",&n,&m,&s,&t);
in[s]++;in[t]--;
int S=0,T=n+1,count=0;
for(int i=0;i<m;i++)
{
scanf("%d%d%d%d",&u,&v,&a,&b);
if(a<=b)
{
in[u]--;in[v]++;
addEdge(v,u,1,b-a);
sum+=a;
}
else
{
addEdge(u,v,1,a-b);
sum+=b;
}
}
for(int i=1;i<=n;i++)
{
if(in[i]>0)addEdge(S,i,in[i],0);
if(in[i]<0)addEdge(i,T,-in[i],0),count+=-in[i];
}
while(SPFA(S,T))
{
end(S,T);
}
printf("Case %d: ",cas++);
if(max_flow==count)printf("%d\n",sum+ans);
else printf("impossible\n");
}
return 0;
}