### Network flow 2 · Maximum flow minimum cut theorem

The time limit :10000ms
Single point time limit :1000ms
Memory limit :256MB

#### describe

Small Hi： In the last week's Hiho In the following part, we will initially explain the concept of network flow and the conventional solution , Small Ho Do you remember the content ？

Small Ho： I remember ！ Network flow is given a graph G=(V,E), And the source s And meeting point t. Every side e(u,v) With capacity c(u,v). The maximum flow problem of network flow is solved from s To t How much traffic can there be at most .

Small Hi： The solution to this problem ？

Small Ho： The basic idea to solve the network flow is to find the augmentation path , Keep updating the residual network . Until we can't find a new way to expand , The flow obtained at this time is the maximum flow of the network .

Small Hi： you 're right , It seems that you remember very well .

Small Ho： Hey, hey, hey , But here I have a problem , Why can't we find Zengguang road and find the maximum flow ？

Small Hi： This time I'll solve your doubts , First of all, let's start with the cut of network flow .

For a network flow graph G=(V,E), Its cut is defined as a way of dividing points ： Divide all the points into S and T=V-S Two parts , Where the source point s∈S, Confluence t∈T.

For a cut (S,T), We define net flow f(S,T) It means to cut through (S,T) The sum of all the traffic , namely ：

`f(S,T) = Σf(u,v) | u∈S,v∈T`

for instance ( This example is from introduction to Algorithms )： ` Net flow f = f(2,4)+f(3,4)+f(3,5) = 12+(-4)+11 = 19`

At the same time, we define the capacity of cuts C(S,T) For all from S To T The sum of the edge capacities of , namely ：

`C(S,T) = Σc(u,v) | u∈S,v∈T`

Also in the example above , Its cutting capacity is ：

`c(2,4)+c(3,5)=12+11=23`

Small Ho： That is to say, we are calculating the cut (S,T) The net flow of f(S,T) There may be a reverse flow that makes f(u,v)<0, And capacity C(S,T) It must be a non negative number .

Small Hi： You're right to say that . In fact, for any cut of the net flow f(S,T) It's always related to the flow of network traffic f equal . For example, in the above example, let's change the way of cutting ： We can calculate the net flow for both cases f(S,T) Still equal to 19.

An intuitive explanation is ： According to the definition of network flow , Only the source s It creates traffic , Confluence t Will receive traffic . So any non s and t The point of u, Its net flow must be 0, That is to say Σ(f(u,v))=0. And the source s All the traffic will eventually pass through the cut (S,T) To the meeting point t, So the flow of network flow f It's equal to the static flow of the cut f(S,T).

The strict proof is as follows ：

```f(S,T) = f(S,V) - f(S,S)
from S To T The flow of is equal to from S The flow to all nodes is subtracted from S To S The flow of internal nodes
f(S,T) = f(S,V)
because S The flow between internal nodes must have corresponding reverse flow , therefore f(S,S)=0
f(S,T) = f(s,V) + f(S-s,V)
then S The set is divided into source points s And others belong to S The node of
f(S,T) = f(s,V)
Because in addition to the source point s No flow is generated by other nodes , therefore f(S-s,V)=0
f(S,T) = f(s,V) = f```

therefore f(S,T) It's from the source s Out of the stream , That's the flow of the network f.

Small Ho： Simple words , That is to say, the net flow of any cut f(S,T) Is equal to the current network traffic f.

Small Hi： That's true . And for the net flow of any cut f(S,T) It must be less than or equal to the capacity of the cut C(S,T). That is to say , For any stream in the network f It must be less than or equal to the capacity of any cut C(S,T).

And in all possible cuts , There is a cut with the smallest capacity , We call it the minimal cut .

This minimal cut limits the flow of a network f upper bound , So there is ：

For any network flow graph , The maximum flow must be less than or equal to the minimum cut .

Small Ho： But what does this have to do with Zengguang road ？

Small Hi： Next is the point . Using the knowledge above , We can deduce a maximum flow minimum cut theorem ：

``` For a network flow graph G=(V,E), One of the active points s And meeting point t, So the following three conditions are equivalent ：
1. flow f The picture is G The maximum flow of
2. Residual network Gf There is no augmentation road
3. about G Some cut of (S,T), here f = C(S,T)```

First of all, prove that 1 => 2：

` We use the counter evidence , Suppose the flow f The picture is G The maximum flow of , But there are still augmented paths in the residual network p, Its flow is fp. Then we have flow f'=f+fp>f. This is related to f It's the maximum flow that leads to conflict .`

And then prove 2 => 3：

``` Suppose the residual network Gf There is no augmentation road , So in the residual network Gf There is no path from s arrive t. We define S The set is ： In the current network s Points that can be reached . At the same time define T=V-S.
here (S,T) Make a cut (S,T). And for any u∈S,v∈T, Yes f(u,v)=c(u,v). if f(u,v)<c(u,v), Then there are Gf(u,v)>0,s It can reach v, And v Belong to T contradiction .
So there is f(S,T)=Σf(u,v)=Σc(u,v)=C(S,T).```

Finally, prove 3 => 1：

` because f The upper bound of is the minimum cut , When f When we reach the capacity of the cut , It's obvious that we've reached the maximum , therefore f For maximum flow .`

This explains why we can't find Zengguang road , It must be the maximum flow .

Small Ho： I see , Oh, I see .

#### Input

The first 1 That's ok ：2 A positive integer N,M.2≤N≤500,1≤M≤20,000.

The first 2..M+1 That's ok ： Each row 3 It's an integer u,v,c(u,v), To represent an edge (u,v) And its capacity c(u,v).1≤u,v≤N,0≤c(u,v)≤100.

The default source point in the given graph is 1, The meeting point is N. There may be duplicate edges .

#### Output

The first 1 That's ok ：2 It's an integer A B,A Represents the capacity of the smallest cut ,B Represents a given graph G Minimum cut S The number of points in the set .

The first 2 That's ok ：B An integer separated by spaces , Express S The point number of the set .

If there are multiple minimal cuts, the solution of any one can be output .

The sample input
```6 7
1 2 3
1 3 5
2 4 1
3 4 2
3 5 3
4 6 4
5 6 2```
Sample output
```5 4
1 2 3 5
analysis ： Min cut max flow ,dicnic; Code ：```
```#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cmath>
#include <algorithm>
#include <climits>
#include <cstring>
#include <string>
#include <set>
#include <map>
#include <queue>
#include <stack>
#include <vector>
#include <list>
#define rep(i,m,n) for(i=m;i<=n;i++)
#define rsp(it,s) for(set<int>::iterator it=s.begin();it!=s.end();it++)
#define mod 1000000007
#define inf 0x3f3f3f3f
#define vi vector<int>
#define pb push_back
#define mp make_pair
#define fi first
#define se second
#define ll long long
#define pi acos(-1.0)
#define pii pair<int,int>
#define Lson L, mid, rt<<1
#define Rson mid+1, R, rt<<1|1
const int maxn=5e2+;
using namespace std;
ll gcd(ll p,ll q){return q==?p:gcd(q,p%q);}
ll qpow(ll p,ll q){ll f=;while(q){if(q&)f=f*p;p=p*p;q>>=;}return f;}
int n,m,k,t,h[maxn],tot,vis[maxn],s,cur[maxn];
bool flag;
set<int>ans;
struct node
{
int to,nxt,cap,flow;
}e[<<];
{
e[tot].to=y;
e[tot].nxt=h[x];
e[tot].cap=z;
h[x]=tot++;
e[tot].to=x;
e[tot].nxt=h[y];
h[y]=tot++;
}
bool bfs()
{
memset(vis,,sizeof vis);
queue<int>p;
p.push(s);
vis[s]=;
if(flag)ans.insert(s);
while(!p.empty())
{
int x=p.front();p.pop();
for(int i=h[x];i!=-;i=e[i].nxt)
{
int to=e[i].to,cap=e[i].cap,flow=e[i].flow;
if(!vis[to]&&cap>flow)
{
vis[to]=vis[x]+;
p.push(to);
if(flag)ans.insert(to);
}
}
}
return vis[t];
}
int dfs(int x,int a)
{
if(x==t||a==)return a;
int ans=,j;
for(int&i=cur[x];i!=-;i=e[i].nxt)
{
int to=e[i].to,cap=e[i].cap,flow=e[i].flow;
if(vis[to]==vis[x]+&&(j=dfs(to,min(a,cap-flow)))>)
{
e[i].flow+=j;
e[i^].flow-=j;
ans+=j;
a-=j;
if(a==)break;
}
}
return ans;
}
int max_flow(int s,int t)
{
int flow=,i;
while(bfs())
{
rep(i,,n)cur[i]=h[i];
flow+=dfs(s,inf);
}
return flow;
}
int main()
{
int i,j;
memset(h,-,sizeof h);
scanf("%d%d",&n,&m);
while(m--)
{
int a,b,c;
scanf("%d%d%d",&a,&b,&c);
}
s=,t=n;
printf("%d",max_flow(s,t));
flag=true;
bfs();
printf(" %d\n",ans.size());
for(int x:ans)printf("%d ",x);
printf("\n");
//system("Pause");
return ;
}```

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