Harry Potter and the Forbidden Forest
Time Limit: 5000/3000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)
Total Submission(s): 1549 Accepted Submission(s): 528
Problem Description
Harry Potter notices some Death Eaters try to slip into Castle. The Death Eaters hide in the most depths of Forbidden Forest. Harry need stop them as soon as.
The Forbidden Forest is mysterious. It consists of N nodes numbered from 0 to N-1. All of Death Eaters stay in the node numbered 0. The position of Castle is node n-1. The nodes connected by some roads. Harry need block some roads by magic and he want to minimize
the cost. But it’s not enough, Harry want to know how many roads are blocked at least.
Input
Input consists of several test cases.
The first line is number of test case.
Each test case, the first line contains two integers n, m, which means the number of nodes and edges of the graph. Each node is numbered 0 to n-1.
Following m lines contains information about edges. Each line has four integers u, v, c, d. The first two integers mean two endpoints of the edges. The third one is cost of block the edge. The fourth one means directed (d = 0) or undirected (d = 1).
Technical Specification
1. 2 <= n <= 1000
2. 0 <= m <= 100000
3. 0 <= u, v <= n-1
4. 0 < c <= 1000000
5. 0 <= d <= 1
Output
For each test case:
Output the case number and the answer of how many roads are blocked at least.
Sample Input
3 4 5 0 1 3 0 0 2 1 0 1 2 1 1 1 3 1 1 2 3 3 1 6 7 0 1 1 0 0 2 1 0 0 3 1 0 1 4 1 0 2 4 1 0 3 5 1 0 4 5 2 0 3 6 0 1 1 0 0 1 2 0 1 1 1 1 1 2 1 0 1 2 1 0 2 1 1 1
Sample Output
Case 1: 3
Case 2: 2
Case 3: 2
分析：题意为求最小割边中边数最小的割的割边数。方法有很多种：
类型一：跑第一遍最大流，把残余网络中的割边（即剩余流量为0的变）赋权值为1，其余为inf，再一遍最大流即可。
类型二：通过改变边的权值，想象如果每一条边容量都加1，那么最小割中，边数多的的割加的流量多于边数少的割的流量，那么边数多的就不是割了（流量打了）
当然不是绝对加一，使边的权值w变为w*（E+1）+1，是最大流%(E+1),此为最小割最小边数，这样防止改变最小割的结果。
代码示例（类型1）：
#include<stdio.h>
#include<string.h>
#include<algorithm>
#include<iostream>
#include<queue>
#define Lh 2000
#define Le 2000000
#define max 1000000000
using namespace std;
typedef struct
{
int to;
int w;
int next;
}node;
typedef struct
{
int x;
int t;
}DEP;
node E[Le];
DEP fir,nex;
int head[Lh],headx[Lh],deep[Lh],cnt;
void init()
{
memset(head,-1,sizeof(head));
cnt=0;
}
void add(int a,int b,int c)
{
E[cnt].to=b;
E[cnt].w=c;
E[cnt].next=head[a];
head[a]=cnt++;
E[cnt].to=a;
E[cnt].w=0;
E[cnt].next=head[b];
head[b]=cnt++;
}
int min(int x,int y)
{
return x<y?x:y;
}
int bfs(int s,int t,int n)
{
memset(deep,255,sizeof(deep));
queue<DEP>Q;
fir.x=s;
fir.t=0;
deep[s]=0;
Q.push(fir);
while(!Q.empty())
{
fir=Q.front();
Q.pop();
for(int i=head[fir.x];i!=-1;i=E[i].next)
{
nex.x=E[i].to;
nex.t=fir.t+1;
if(deep[nex.x]!=-1||!E[i].w)
continue;
deep[nex.x]=nex.t;
Q.push(nex);
}
}
for(int i=0;i<=n;i++)
headx[i]=head[i];
return deep[t]!=-1;
}
int dfs(int s,int t,int flow)
{
if(s==t)
return flow;
int newflow=0;
for(int i=headx[s];i!=-1;i=E[i].next)
{
headx[s]=i;
int to=E[i].to;
int w=E[i].w;
if(!w||deep[to]!=deep[s]+1)
continue;
int temp=dfs(to,t,min(w,flow-newflow));
newflow+=temp;
E[i].w-=temp;
E[i^1].w+=temp;
if(newflow==flow)
break;
}
if(!newflow)deep[s]=0;
return newflow;
}
int Dinic(int s,int t,int m)
{
int sum=0;
while(bfs(s,t,m))
{
sum+=dfs(s,t,max);
}
return sum;
}
int main()
{
int a,b,c,d;
int n,m,T;
scanf("%d",&T);
for(int t=1;t<=T;t++)
{
scanf("%d%d",&n,&m);
init();
for(int i=0;i<m;i++)
{
scanf("%d%d%d%d",&a,&b,&c,&d);
add(a,b,c);
if(d)
add(b,a,c);
}
Dinic(0,n-1,n);
for(int i=0;i<=cnt;i+=2)
{
if(!E[i].w)
E[i].w=1;
else
E[i].w=max;
}
printf("Case %d: %d\n",t,Dinic(0,n-1,n));
}
return 0;
}