Problem:
Score : 600 points

Problem Statement

Kenkoooo found a simple connected graph.
The vertices are numbered 1 through n.
The i-th edge connects Vertex u_i and v_i, and has a fixed integer s_i.

Kenkoooo is trying to write a positive integer in each vertex so that the following condition is satisfied:

For every edge i, the sum of the positive integers written in Vertex u_i and v_i is equal to s_i.

Find the number of such ways to write positive integers in the vertices.

Constraints

2 \leq n \leq 10^5

1 \leq m \leq 10^5

1 \leq u_i < v_i \leq n

2 \leq s_i \leq 10^9

If i\neq j, then u_i \neq u_j or v_i \neq v_j.

The graph is connected.

All values in input are integers.

Input

Input is given from Standard Input in the following format:

n m
u_1 v_1 s_1
:
u_m v_m s_m

Output

Print the number of ways to write positive integers in the vertices so that the condition is satisfied.

Sample Input 1

3 3
1 2 3
2 3 5
1 3 4

Sample Output 1

1

The condition will be satisfied if we write 1,2 and 3 in vertices 1,2 and 3, respectively.
There is no other way to satisfy the condition, so the answer is 1.

Sample Input 2

4 3
1 2 6
2 3 7
3 4 5

Sample Output 2

3

Let a,b,c and d be the numbers to write in vertices 1,2,3 and 4, respectively.
There are three quadruples (a,b,c,d) that satisfy the condition:

(a,b,c,d)=(1,5,2,3)

(a,b,c,d)=(2,4,3,2)

(a,b,c,d)=(3,3,4,1)

Sample Input 3

8 7
1 2 1000000000
2 3 2
3 4 1000000000
4 5 2
5 6 1000000000
6 7 2
7 8 1000000000

Sample Output 3

0

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include<iostream>
#include<algorithm>
#include<vector>
using namespace std;
typedef long long ll;
typedef pair<ll,ll> P;
#define int ll
vector<P> node[100005];
ll num[100005]={0},two_j[100005]={0};
bool used[100005]={false};
void dfs(ll pos,ll k){
  two_j[pos]=k;
  k%=2; k++;
  for(int i=0;i<node[pos].size();i++){
    ll to=node[pos][i].first;
    if(two_j[to]==0){
      two_j[to]=k;
      dfs(to,k);
    }
  }
}
void set_val(ll pos,ll val){
  num[pos]=val;
  for(int i=0;i<node[pos].size();i++){
    ll to=node[pos][i].first,cos=node[pos][i].second;
    if(!used[to]){
      used[to]=true;
      set_val(to,cos-val);
    }
  }
}
signed main(){
  ll n,m; cin>>n>>m;
  for(int i=0;i<m;i++){
    ll a,b,c; cin>>a>>b>>c;
    node[a].push_back(P(b,c));
    node[b].push_back(P(a,c));
  }
  dfs(1,1);
  for(int i=1;i<=n;i++) used[i]=false;
  used[1]=true;
  set_val(1,1);

  bool f=false;
  for(int i=1;i<=n;i++){
    for(int j=0;j<node[i].size();j++){
      ll to=node[i][j].first;
      if(two_j[i]==two_j[to]) f=true;
    }
  }
  if(f){
    for(int i=1;i<=n;i++){
      for(int j=0;j<node[i].size();j++){
        ll to=node[i][j].first,cos=node[i][j].second;
        if(num[i]+num[to]!=cos){
          ll sa=cos-num[i]-num[to];
          sa/=2;
          for(int l=1;l<=n;l++){
            num[l]+=sa;
            if(num[l]<=0){
              cout<<0<<endl;
              return 0;
            }
            else{
              cout<<1<<endl;
              return 0;
            }
          }
        }
      }
    }
    cout<<1<<endl;
    return 0;
  }
  else{
    ll one_m=174417441744174417,two_m=174417441744174417;
    for(int i=1;i<=n;i++){
      if(two_j[i]==1) one_m=min(one_m,num[i]);
      else two_m=min(two_m,num[i]);
    }
    cout<<max((ll)0,(one_m+two_m-1))<<endl;
  }
}